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About Google Book Search Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web at |http: //books .google .com/I BIBLIOGRAPHIC RECORD TARGET Graduate Library University of Michigan Preservation Office Storage Number: ACQ7946 UL FMT B RT a BL m T/C DT 07/18/88 R/DT 07/18/88 CC STATmmE/Ll 010: : I a 05028039 035/1: : ] a (RLIN)MIUG86-B28757 035/2: : j a (CaOTULAS)160537099 040: : | a MnU | c MnU | d MiU 050/1:0: |aQA371 |b.F7 100:1 : I a Forsyth, Andrew Russell, | d 1858-1942. 245:00: | a Theory of differential equations. | c By Andrew Russell Forsyth. 260: : | a Cambridge, [ b University Press, jcl890-1906. 300/1: : la4pts. in6v. |c23cm. 505/1:1 : | a pt I. (Vol. I) Exact equations and Pfaff s problem. 1890.-pt. 11. (Vol. II-lII) Ordinary equations, not linear. 1900.-pt. III. (Vol. IV) Ordinary equations. 1902.--pt. IV (vol. V-VI) Partial differential equations. 1906. 590/2: : |aastr: Pt.l (vol 1) only 650/1:0: | a Differential equations 998: : | c DPJ | s 9124 Scanned by Imagenes Digitales Nogales, AZ On behalf of Preservation Division The University of Michigan Libraries Date work Began: _ Camera Operator: _ y Google THEORY DIFFERENTIAL EQUATIONS. y Google aouOon; 0. J. CLAY and SONS, CAMBKIDGE UNIVERSITY PEESS WAREHOUSE, LANE, NGTON STREET, m: F. A. EROCKHAUS. TUE MAOMILLAN COMPANY. ciilfa: MACMILLAN AND 00., Lru. y Google THEOEY OF DIFFEEENTIAL EQUATIONS. PAUT III. OKDINAKY LINEAR EQOATIONS. ANDREW EUSSELL FOBSYTH, ScD., LL.D., F.K.S., CAMBRIDGE: AT THE UNIVERSITY PRESS. 1902 All rights r^ssrued. y Google y Google PBEPACE. The present volume, constituting Part III of this work, deals with the theory of ordinary linear differential equations. The whole range of that theory is too vast to be covered by a single volume ; and it contains several distinct regions that have no organic relation with one another. Accordingly, I have limited the discussion to the single region specially occupied by applications of the theory of functions ; in imposing this limitation, my wish has been to secure a uniform presentation of the subject. As a natural consequence, much is omitted that would have been included, had my decision permitted the devotion of greater space to the subject. Thus the formal theory, in its various shapes, is not expounded, save as to a few topics that arise incidentally in the functional theory. The association with homogeneous forms is indicated only slightly. The discussion of com- binations of the coefficients, which are invariautive under all transformations that leave the equation linear, of the associated equations that are covaviantive under these transformations, and of the significance of these invariants y Google and covariants, is completely omitted. Nor is any appli- cation of the theory of groups, save in a single functional investigation, given here. The student, who wishes to consider these subjects, and others that have been passed by, will find them in Schlesinger's Handhuch der Theorie der linearen Differentialgleichungen, in treatises such as Pieard's Corns d'Analyse, and in many of the memoirs quoted in the present volume. In preparing the volume, I have derived assistance from the two works just mentioned, as well as from the uncompleted work by the late Dr Thomas Craig. But, as will be seen from the references in the text, my main assistance has been drawn from the numerous memoirs contributed to learned journals by various pioneers in the gradual development of the subject. Within the limitations that have been imposed, it will be seen that much the greater part of the volume is assigned to the theory of equations which have uniform coefficients. When coefficients are not uniform, the difficulties in the discussion are grave : the principal characteristics of the integrals of such an equation have, as yet, received only slight elucidation. On this score, it will be sufficient to mention equations having algebraic coefficients : nearly all the characteristic results that have been obtained are of the nature of existence-theorems, and little progress in the difficult task of constructing explicit results has been made. Moreover, I have dealt mainly witli the general theory and have abstained from developing detailed properties of the functions defined by important par- ticular equations. The latter have been used as illustra- tions ; had they been developed in fuller detail than is y Google given, the investigations would soon have merged into discussions of the properties of special ■ functions. In- stances of such transition are provided in the functions, defined by the hypergeometric equation and by the modern form of Lamp's equation respectively. A brief summary of the contents will indicate the actual range of the volume, In the first Chapter, the synectic integrals of a linear equation, and the conditions of their uniqueness, are investigated. The second Chapter discusses the general character of a complete system of integrals near a singularity of the equation. Chapters III, IV, and V are concerned with equations, which have their integrals of the type called regular ; in particular. Chapter V contains those equations the integrals of which are algebraic functions of the variable. In Chapter VI, equations are considered which have only some of their integrals of the I'egular type ; the influence of such integrals upon the reducibility of their equation is in- dicated. Chapter VII is occupied with the determination of integrals which, whUe not regular, are irregular of specified types called normal and subnormal ; the functional significance of such integrals is established, in connection with Poincare's development of Laplace's solution in the form of a definite integral. Chapter VIII is devoted to equations, the integrals of which do not belong to any of the preceding types ; the method of converging infinite determinants is used to obtain the complete solution for any such equation. Chapter IX relates to those equations, the coefficients of which are uniform periodic functions of the variable : there are two y Google classes, according as the periodicity is simple or double. The final Chapter deals with equations having algebraic coefficients; it contains a brief genera! sketch of Poincare's association of such equations with automorphic functions. In the revision of the proof-sheets, 1 have received valuable assistance from three of my friends and former pupils, Mr. E. T. Whittaker,M.A., and Mr. E. W. Barnes, M.A., Fellows of Trinity College, Cambridge, and Mr, E. W. H. T. Hudson, M.A., Fellow of St John's College, Cambridge ; I grateftilly acknowledge the help which they have given me. And I cannot omit the expression of my thanks to the Staff of the University Press, for the unfailing courtesy and readiness with which they have lightened my task during the printing of the volume. A. R. FORSYTH. Trisity Cult.bge, Cambeidgb, 1 jtfareA, 1902. y Google CONTENTS. CHAPTER I. LINEAR EQUATIONS : EXISTENCE OF SYNECTIC INTEGRALS ; FUNDAMENTAL SYSTEMS. Introductory remarks Form of the homogeneous linear equation of oidei m Eatablisbmeut of the esistence of a sjneutii, integrril in the domeiin of an ordinary puint, dotennined uniquely by the initial conditions with corollanea, and Bsamples Hermite's treatment of the equation with constant tooflicienta Continuation of the aynectie integral beyond the initial domain ; r^on of its continuity bounded by the singu- larities of the equation Certain deformations of path of independent variable leave the final int^ral unchanged ...... Seta of integrals determined by sets of initial valuea . The determinant ^ {z) as affecting the linear independence of a sot of m integials a fundamental system and the eSective test of it* fitness Any integial is linearly e:£press!ble in terms of the elements of a fundlmentul sjstem ..... Construction if i speiiil fundamental systeni CHAPTER II. GENERAL FORH AND PROPERTIES OP INTEGRALS NEAR A SINGULARITY. 13. Constructitn of the fundwmentcd equation belonging to a singularity 35 14. The fundimental equation is independent of the choice of the f\indimcntal aytera Poincarii'a theorem ... 38 y Google CONTENTS Tl 1 m tay d &ors f tt f d t ! q it la tal f m L n T y 13 p joat w b 11 4. fi d m ta] y t f t^r 1 1: th f dm tal ©q tat diat t f Eff t f m Itpl oot T fmt fthfdmtleqt 1 f th d ced f n l_ 1 f teiral ted with Itil fth g p t bgi [ TI bmg b g 1 eq t OB h t tegral t g" P d th g i t th fegnl ■y I hoat f th Ijt 1 I f th Hfti b b-g p ult th teg al 1 IM f 1 relating to such integrals . f tl lyt I CHAPTER III. RRQULAR ISTEGBALS 1 EQUATION HAVING ALL ITS INTEGRALS RBRTJI.AI NEAR A SINGULARITY. 39. Definition of integral, regular in the vicinity of a singularity : index to which it belongs 7 30. Iddes of the dctemiioant of a fundamental system of integrals all of which are regular near the singularity ... 7 31. Form of homogeneous linear equation when all its integrals are regular near a 7 32. 33. Converse of the preceding reault, established by the method of Frobenius ......... 7 34. Series proved to converge uniformly ajid unconditionally . 8 35, Integral associated with a simple root of an algebraic (in- dicia!) equation ..,.■... i 36—38. Sot of int^rals associated with special group of roots of the algebraic (indicial) equation, with summaj'y of results when all the integrals are regular E 39. Definition of indieial equation, indidol funetiiyii : signiiicance of integrals obtained i 40. The iiit^rals obtained constitute a fundamental system ; with y Google Conditiona that ever3 iPgular integral belonging to a par- ticulir exiionent '.hould have its espression free from logiiithms , with es^mple'J ...... Condititni that there should be at least one r^ular integral belonging to a partitulai exponent and free from loga- iithms ...... Alternitue method aometimea effective for settling the ques- tion in §§ 43, 43 Discnmination between real singularity and apparent singu- lant; conditions fot the latter ..... Noh on thp leties in 1) S4 CHAPTER IV, EQUATIONS HAVING THEIR INTEGKALS REGULAR IN THE VICINITY OF EVERY SINGULARITY (iKCLOUING infinity). 46. Equations (said to be of the FacAsian type) having all their int^r&ls r^ular in the vicinity of every singularity {including =o ) ; their form : with examples . . . 123 47. Equation of second order completely determined by assign- ment of singularities and tlieir exponents : Riemaun's jp-function 135 48. Significance of the relation among the exponents of the pre- ceding equation and fimotion 139 49. Oonatruction of the differential equations thus determined . 141 50. The equation satisfied by the hype^eometric series, with 51. 52. Equations of the Fuclisian type and the second order with more than three singularities (i) when co is not a singu- larity, (ii) when oo is a singularity ..... 150 53. Normal forme of such equations 156 54. Lamp's equation transformed so as to be of Fuchsiaa type . 160 55. ■ B6clier'a theorem on the relation between the linear equations of mathematical physics and an equation of the second order and Fuchaian type with five singularities . . 161 56. Heine's equations of the second order having an integral that is a polynomial 165 57. Equations of the second order all whose integrals are rational 169 y Google CHAPTER V. 1 KQUATIONS OP THE SECOND AKD THE THIRD OEDERS i ALGEBRAIC I Mutlnj la of letermining whether an equation hi^ dlt,pl i ii ntegi ill Kle a a special meth id for detonnining all the finite groups fc thp equation of the ^uond ordei Ihp pq ationa satisfied by the quotient of two 'olution-j the equition of the second order then integrals L m'iti-urtion of equations of the second order a" ntegrable Meini of deteiminmg whether a gi\eii equation is n braicaily uitegralle with examples Lqu-itims jf the third order their quotient eq lat ons Painleve >■ invariants, cirreaponding to the Scliwaiziin der tive for the equation of the second order connet with La^uerres mvanant A'.i loiation with finite group? of trai aformat oi 1 neo-linear in two variables IndiLitims Df other poMible methods Eemiik-J on equatmn^ of the foiuth. orler Association of equations of the third ani highei triers the theon of homogeneous forms And of equati >ns of the second order "Diacuasion of equations of the thiri order with i -"e theorem due to Fuch-i with cs'implp in I roferei < e j^uations of higher order th^t ' CHAPTER VI. lUA'l'IONS HAVING ONLY SOME OF THEIR INTEGHALS REGULAR NEAR A SINGULARITY. Equations having only some of their integrals regular ii vicinity of a singularity : the characteristic indes . The linearly independent aggregate of r^ular integrals satisfy a liiieaj equation of order equal to their number , Reducible equations Frobenius's characteristic function, indicial function, ivdiciid eqwitwji ; normal form of a differential equation associ able with the indicia] function, uniquely determined b; the characteristic function The number of r^ular iutegrals of an equation of order ii and oharaoterietic index n is not greater than m — n y Google CONTENTS Tlr mbe f gul tp ! be 1 sa th D te t f tl regul t 1 h tii y t w th mpl E t f irred bl equat Aqt fil h g dpedt ulor te b las grl t 1 a. tedw th it f rd La eqtndf to^n qt Eelt htee qt dtadit, pett f th L fheil Ipdtrel t orals Jia dhlt I t CHAPTER Vn. NORMAL integrals: suenosmal integrals. Integrals for which the singularity of the equation is essential : normal int^rals Thome's method of obtaining norma! integrals when they Construction of determining factor : possible cases . Svhnormal integrals Rank of an equation ; Poincar^'s theorem od a set of normal or subnormal functions as integrals ; examples Hftmbu:^er's equations, having s=0 for an essential singu- larity of the integrale, which are regular at tc and elsewhere are sjnectic : equation of second order Cayley's method of obtaining normal iut^rals . Hamburger's equations of order higher than the second Conditions associated with a simple root of the character- istic equation for the determining factor . Likewise for a multiple root Subnormal int^rals of Hamburger's equations Detailed discussion of equation of the third order . Normal integrals of equations with rational coefficients Poincar^s development of Laplace's solution for grad( 1. Liapouuoff's theorem ....... 1—105. Application to the evaluation of the definite int^ral ir Laplace's solution, leading to a normal integral i. Double-loop integrals, after Jordan and Pochhammer '. When the normal series diverges, it is an asymptotic repre- sentation of the definite-integral solution i. PoincarS's transformation of equations of rank higher than unity to equations of rank unity y Google CHAPTER V1.IT. INFINITE DETERMISANTS, AND THEIK APPLICATION TO THE SOLUTION OF LINBAU EQUATIONS. 109. lutljluU 311 Ji iniimte determnints tLsts of c l\OY- s;cnci, 1 io[>eities . 348 110. Dciebitiicnt . 352 111. Mm I'l 353 112. IT[ ftrm comergence when t iistitui.nt-< aie fun t ons of ii. paiametei . 358 113. Solution of an unlimited r umbct of simultaneous In ear equations . 360 114. Difletential equations having no regular totegial oo noimal integial no subnormal integtal 363 115. Inf-egral in the form (f a Laurent aenes intioducticn of !iii inhmte determinant ii{pj ■ SGii IIG. C nvergent* ot Q(p) . 366 117. Introiuction of i^nother inimite deteiminint I>{p) its convergente and its telation to Q{p) with deduced eiiieSBion of £2(p) . 369 118. ( niei^ence of tlie Liurent ^nea exj reding the mtegial . 376 119. & neiihsation of mpthod ol Fiobennis (in Ohap Hi) to determine a system cf mtBgrah . 379 120 — 123. 'V annua cases atoordmg t) the chaiatter of the ineduLiblc r ots of i>{p) = . 380 124. The i^stem ot integrals i5 fundamental . 387 126. Th equati n l>{p)=0 is efiectneh the fundaineatal equa- tion for the combiiidtnn of Bingiilantiei within the cu'cle I |=fi . 389 126. (.eneiil remark example'. . 392 127. Other methc h of ibtaining the fundamental equati ii to which D (fi)=0 IS efiectively equivalent with an example . 398 CHAPTER IX. EQUATIONS WITH UNIFOHII PLRIODIC f DLIf ICIENTS. Equation-fl with mmply 'penodu, coefhcients the funda- mental er[uation a sooiated with the juniod Siiiple looti f the fund^mentil equat on \. iQultii le 1 not of the fundamental equiti ti . y Google 131. Analytical form of tte integrals associated with a root . 411 132. Modification of the form of the group of int^rals associated with a multiple root 414 133. Use of elementary divisors : resolution of group into sub- groups : numbei' of integrals, that are periodic of the second kind 416 134. More precise establiahment of results in § 133 . . . 417 135. Converse proposition, analogous to Fuchs's theorem in § 25 420 136. Further determination of the integrals, with examples . 421 137. Liapounoflfs method 425 138 — 140. Discussion of the equation of the elhptic cylinder, v/' + {a-i-ocOB^)w=^0 . . . .431 141. Equations with doubly-periodic coefficients; the funda- mental equations associated with the periods . . 441 143, 143. Picard's theorem that such an equation poasessea an int^ral which is doubly- periodic of the second kind : the number of such integrals 447 144, 145. The int^rals associated with multiple roots of the funda- mental equations ; two cases ..... 451 146. First stage in the construction of analytical expressions of integrals ......... 457 147. Equations that have uniform integrals : with examples . 459 148. Lamp's equation, in the form vf=w{n{n + \)i^{z)-^B), deduced from the equation for the potential . . 464 149 — 151. Two modes of constructing the integral of Lame's equation 4li8 CHAPTER X. EQUATIONS HAVING ALGEBRAIC COEFFICIENTS. 152. Equations with algebraic coefficients 478 153, 154. Fundamental equation for a singularity, and fundamental systems; examples 480 155, 156. Introduction of automorphic functions .... 498 167, 158. Automorphic property and oonformal representation . 491 159—161. Automorphic property and linear equations of second order 495 162. Illustration from elliptic functions 501 163. Equations with one singularity 506 184. Equations with two singularities 507 165. Equations with three singularities 508 166. General statement as to equations with any number of singularities, whether real or complex . . . BIO y Google statement of Poiucar^'s results Poincar^'s theorem that any linear equation with algebraic coefficients can be integrated by Fucbsian and Zeta- fucheian functions Properties of these functions : and verification of Poincar^'s theorem Concluding remarks IN BBS I y Google CHAPTER I. Linear Equations ; Existence of Synectic Integrals : Fundamental Systems. 1. The course of the preceding investigations has made it manifest that the discussion of the properties of functions, which are defined by ordinary differential equations of a general t3rpe, rapidly increases in difficulty with successive increase in the order of the equations. Indeed, a stage is soon reached where the generality of form permits the deduction of no more than the simplest properties of the functions. Special forms of equations can be subjected to special treatment ; but, when such special forms conserve any element of generality, complexity and difficulty arise for equations of any but the lowest orders. There is one exception to this broad statement ; it is constituted by ordinary equations which are hnear in form. They can be treated, if not in complete generality, yet with sufficient fulness to justify their separate discussion ; and accordingly, the various important results relating to the theory of ordinary linear differential equations constitute the subject-matter of the present Part of this Treatise. Some classes of linear equations have received substantial consideration in the construction of the customary practical methods used in finding solutions. One particular class is com- posed of those equations which have constants as the coefficients of the dependent variable and its derivatives. There are, further, equations associated with particular names, such as Legendre, Bessel, Lam^ ; there are special equations, such as those of the hypergeometric series and of the quarter-period in the Jacobian theory of elliptic functions. The formal solutions of such equations y Google 2 HOMOGENEOUS [1 . can be regarded as known; but so long as the investigation is restricted to the practical construction of the respective series adopted for the solutions, no indication of the range, over which the deduced solution is valid, is thereby given. Ifc is the aim of the general theory, as applied to such equations, to reconstruct the various methods of proceeding to a solution, and to shew why the isolated rules, that seem so sourceless in practice, actually prove effective. In prosecuting this aim, it will be necessary to revise for linear equations all the customarily accepted results, so as to indicate their foundation, their range of validity, and their signiiicance. For the most part, the equations considered will be kept as general as possible within the character assigned to them. But from titne to time, equations will be discussed, the functions defined by which can be expressed in terms of functions already known ; such instances, however, being used chiefly as illustrations. For all equations, it will be necessary to consider the same set of problems as present themselves for consideration in the discussion of unrestricted ordinary equations of the lowest orders : the exist- ence of an integral, its uniqueness as determined by assigned conditions, its range of existence, its singularities (as regards position and nature), its behaviour in the vicinity of any singu- larity, and so on : together with the converse investigation of the limitations to be imposed upon the form of the equation in order to secure that functions of specified classes or types may be solutions. As is usual in discussions of this kind, the variables and the parameters will he assumed to be complex. It is true that, for many of the simpler applications to mechanics and physics, the variables and the parametei'S are purely real ; but this is not the case with all such applications, and instances occur in which the characteristic equations possess imaginary or complex parametei-s or variables. Quite independently of thk latter fact, however, it is desirable to use complex variables in order to exhibit the proper i-elation of functional variation. 2. Let z denote the independent variable, and w the dependent variable ; z and w varying each in its own plane. The differential equation is considered linear, when it contains no term of order higher than the first in w and its derivatives ; and a linear equation is called homogeneous, when it contains no term independent of w y Google 2.] LINEAR EQUATIONS 3 and its derivatives. By a well-known formal result*, the solution of an equation that is not homogeneous can be deduced, merely by quadratures, from the solution of the equation rendered homo- geneous by the omission of the term independent of w and its derivatives ; .and therefore it is sufficient, for the purposes of the general investigation, to discuss homogeneous linear equations. The coefficients may be uniform functions of s. either rational or transcendental ; or they may be multiform functions of a, the simplest instance being that in which they are of a form (s, z), where is rational in s and z, and s is an algebraic function of z. Examples of each of these classes will be considered in turn. The coefficients will have singularities and (it may be) critical points ; all of these are determinable for a given equation by inspection, being fixed points which are not affected by any constants that may arise in the integration. Such points will be found to include all the singularities and the critical points of the integrals of the equation ; in consequence, they are frequently called the singu- larities of the equation. Accordingly, the differential equation, assumed to be of order m, can be taken in the form where the coefficients p^, p^, ..., pm are functions of s. In the earlier investigations, and until explicit statement to the contrary is made, it will be assumed that these functions of z are uniform within the domain considered ; that then' singularities are isolated points, so that any finite part of the plane contains only a limited number of them : and that all these singularities (if any) for finite values of s are poles of the coefficients, so that their only essential singularity (if any) must be at infinity. Let f denote any point in the plane which is ordinary for all the coefficients p ; and let a domain of ^ be constructed by taking all the points z in the plane, such that |2-fi«i«-fi. where a is the nearest to ^ amohg all the singularities of all the coefficients. Then within this domain (but not on its boundary) we have P.-P.i'-t). (»-1.2 -»). * See mj Treathte on Differential Eqnatiinui, § 75. y Google 4 SYNECTIC [2. where P„ denotes a regular function o{ s — ^, which generajly is an infinite series of powers of z — f converging within the domain of ^. An integral of the equation existing in this domain is uniquely settled hy the following theorem ; — In the domain of an ordinary point ^, the differential equation possesses an integral, which is a regular function of z — ^ and, with its first m — 1 derivatives, acquires arbitrarily assigned values when z = Z; and this integral is the only regular function of z—^ in the specified domain, which satisfies the equation and fulfils the assigned conditions*. The integral thus obtained will be calledf the synectio integral. Synectic Integrals. §. The existence of an integral which is a holomorphic function of s— ^ within the domain will first be established. Let r he the radius of the domain of i^; let M,, ..., M^ denote quantities not less than the maximum values of \p,], ..., \pm\ respectively, for points within the domain ; and let dominant ictions ^, ... , <|ini, defined by the expressions constructed. Then* for every positive integer a. The dominant functions ^ are used to construct a dominant equation ^ = ■ft S^S=r + <^ rf^S^ + - + ■^'"«. which is considered concurrently with the given equation, * The conditions, as to the arbitrarily assigned values to be ftotiuired at f by tu and itfl derivatives, are called the initial conditions ; the values are called the initial values, t As it is a regular function of the variable, it would have been proper to call it the regular int^cal. This term has however been appropriated [sec Chapter iii, § 39) to describe another class of integrals of linear equations; as the use in this other conneciion is now widespread, oonfusion would result if the use were changed. J See mj Theory of Functions, 2aA edn,, §22; quoted hereafter as ?'. J''. y Google 3,] INTEGRALS 5 Any function which is regular in the domain of ^ can be expressed as a converging series of powers of ^~f; and the coefficients, save as to numerical factors, are the values of the various derivatives of the function at f Accordingly, if there is an integral w which is a regular function of 5 — J^, it can be formed when the values of all the derivatives of w at £f are known. To w. -r- , .--, ~, , the arbitrary values specified in the initial conditions are assigned. All the succeeding derivatives of w can be deduced from the differential equation in the form " rf^""" I- A^,^', (for a — m, m + 1, ... ad inf.), by processes of differentiation, addition, and multiplication: as the coefficient of the highest derivative of w in the equation (and in every equation deduced from it by differentiation) is unity, new critical points are not introduced by these processes, so that all the coefficients A are regular within the domain of f. The successive derivatives of u are similarly expressible in the (for a = m, m + 1, ... ad inf.), obtained in the same way as the equation for the derivatives of w. The coefficients B have the same form as the coefficients A, and can be deduced from them by changing the quantities p and their derivatives into the quantities tp and their derivatives respectively. The values of the derivatives of w and u a,t ^ are required. When i^ = J", all the terms in each quantity B are positive ; on account of the relation between the derivatives of the quantities p and <b, it follows that -B«>|^«|, (s = l, ...,m), .... , „, , \dw\ I d'^'^w I , when 2 = f. Let the mitial values oi |wl, -j- . ■■-, . m^' > when z = ^, be assigned as the values of u, f ,■■■• -j^i^i when 2 = ^; then y Google 6 EXISTENCE OF when«- r. for the valu Mm, m + 1, ... oi :». If the , series (») + (2- /A.\ (i -0- "Ui + 2! converge! -"© denotes the v.,.e„f^'. i'hen z series where ( -r— 1 denotes the value of -^— when z=t, also converses ; Vets"/ ffla" ° it then represents a regular function oi z~ ^ which, after the mode of formation of its coefficients, satisfies the differential equation. We therefore proceed to consider the convergence of the series for u, obtained as a purely formal solution of the dominant equa- tion. To obtain explicit expressions for the various coefficients in this series, let s — f = rw, taking x as the new independent variable. Points within the domain of f are given by |a;|< 1 ; and the dominant equation becomes ax™ s=i dx"^' When the .series for u, taken in the form is substituted in the equation which then becomes an identity, a comparLsoQ of the coefficients of a^ on the two sides leads to the relation holding for all positive integer values of k. This relation shews that all the coefficients h are expressible linearly and homogeneously in terms of 6o, 6i, ..., hw-i- and that, as the first m of these coefficients have been made equal to the moduli of the in arbitrary quantities in the initial conditions for w and therefore are positive, all the coefficients h are positive. Hence k + M,r, yGoosle S.] A SYNECTIC INTEGRAL 7 By the initial definition of M,, it was taken to be not less than the maximum value of \pi\ within the domain of f; it can there- fore be chosen so as to secure that M^r > m. Assuming this choice made, we then have OjK+l > Om+i;— 1 > so that the successive coefficients From the difference- equation satisfied by the coefficients b, it follows that V+*-. k + ni ,^2 (m + k)l ' t™+ft_,' So far as regards the m — 1 terms in the summation, the ratio ^m+ii-s -^ i'm+ft-i is less than unity for each of them ; Mgi^ is finite for each of them; and \in + k — s)\-^{m+k)\ is zero for each of them, in the limit when k is marie infinite. Hence we have and iherefoi'e <1. for points within the domain of f, so that* the series converges within the domain of f. The convergence is not estab- lished for the boundary, so that it can be affirmed only for points within the domain; it holds for all arbitrary positive values assigned to b^, b^, ..., b^-i. It therefore follows that, at all points within the domain of ^, a regular function of s — i^ exists which satisfies the original differential equation for tv, and, with its first m — 1 derivatives, acquires at f arbitrarily assigned values. 4. Now that the existence of a synectic integral is established, the explicit expression of the integral in the form of a power-series in z — ^, this series being known to converge, can be obtained " Cbrystal's Algelra, vol. ii, p. 121. y Google 8 UNIQUENESS OF [4. directly from tlie equation. As f is an ordinary point for each of the coefficients p, we have p,^PA.^-0, (s = l, 2, ...,m), where Pg denotes a regular function of a — f. Let a^, oli, ..., ow-, be the arbitrary values assigned to w, -j— , ..., -j ^~ , when s = f ; and take which manifestly satisfies the initial conditions. In order that this may satisfy the equation, it must make the equation an identity when the expression is substituted therein. When the substitution is effected, and the coefficients of {e — ^y on the two sides of the identity are equated, we have a relation of the form where A^+s is a linear homogeneous function of the coefficients a,, such that K<m + s, and is also linear in the coefficients in the quantities P, (3— f), ..., P„(2 — f); and the relation is valid for s= 0, 1, 2, ..., ad inf. Using the relation for these values of s in succession, we find a^, a^+i, Om+a- ■■■ expressed (in each instance, after substitution of the values of the coeiScients which belong to earlier values of s) as a linear homogeneous function of the quanti- ties Oj, «!, ..., ctnt-ii and in am+s, the expressions, of which the initial constants a^, a^, ..., 0^-1 are coefficients, are polynomials of degree s + 1 in the coefficients of the functions P,{s — ^), .... Pm (^ - ^)- The earlier investigation shews that the power-series for w converges ; accordingly, the determination of the coefficients a in this manner leads to the formal expression of an integral w satisfying the equation. 5, Further, the integral thus obtained is the only regular function, which is a solution of the equation and satisfies the initial conditions associated with a^, eij, ..., 0^-1. If it were possible to have any other regular function, which also is a solu- tion and satisfies the same initial conditions, its expression would be of the form y Google 5.] THE STNECTIC INTEGRAL 9 a regular function oi z — i^. The coefficients would be determin- able, as before, fi'om a relation whore ^'m+j is the same function of a^, ..., a^-i, «'m. ■■-, <t'ni+a-i as ^m+s is of Oo, ..., a^-i, «,„, ..., am+E_i- Hence a'jii+i = -^'in+1 = -dm+i> after substitution for a'^, and so on, in succession. The coefficients agree, and the two series are the same, so that w = w' ', and therefore the initial con- ditions uniquely determine an integral of the equation, which is a regular function of ^ — f in the domain of the ordinary point ?. Corollary I. If all the initial constants ag, a,, ..., a^-i are zero, then the synectic integral of the equation is identically zero. For in the preceding discussion it has been proved that o-m+e, for all the values of s, is a linear homogeneous function of a^, ..., Oot-i ; hence, in the circumstances contemplated, a,n+s = for all the values of s. Thus every coefficient in the series vanishes ; accordingly, the integral is an identical zeio. CoEOLLARY II. The initial constants a^, a^, ..., am_i occur linearly in the ea^ession of the synectic integral ; and each of the m variable quantities, which have those constants /or coe^cients, is a synectic integral of the equation. The first part is evident, because all the coefficients in w are linear and homogeneous in OoiO:,, ..., cW-i. As regards the second part, the variable quantity multiplied by Sg is derivable fi'om w by making a^ = 1, and all the other constants a equal to zero ; these constitute a particular set of initial values which, according to the theorem, determine a synectic integral of the equation. Thus the synectic integral, determined by the initial values a^, ..., "m-i, is of the form aoUi + aiU^+ ... -i-am_iJfm, where each of the quantities m,, u^, ...,«„ is a synectic integral of the equation. j!fote 1. The series of powers oi z — ^, which represents the synectic integral, has been proved to converge within the domain y Google 10 EXISTENCE OF [5. of Z, SO that its radius of convergence is | « - f | , where a is the singularity of the coefficients which is nearest to f. All these singularities lying in the 6nibe part of the plane are determinable by mere inspection of the forms of the coefficients : another method must be adopted in order to take account of a possible singularity when z = co because, even though a = oo may be aji ordinary point of the coefficients, infinite values of the variable affect the character of w and its derivatives. For this purpose, we may change the variable by the substi- tution ea:= 1, and we then consider the relation of the a'-origin to the trans- formed equation as a possible singularity. The transformation of the equation is immediately obtained by means of the formula #w , ,,t * fc ! (fe - 1) ! a^+° d'w^ dz" ^ ^ ,Z^aL\{a.-\)\{k-a)\ dx-' inspection of the transformed equation then shews whether x = () is, or is not, a singularity. Or, without changing the independent variable, we may consider a series for w in descending powers of z : pies will occur hereafter. It may happen that there is no singularity of the coefficients in the finite part of the plane, infinite values then providing the only singularity. In that case, we should not take the quantity r in the preceding investigation as equal to [co — ^], that is, as infinite ; it would suffice that r should be finite, though as large as we please. It may happen that there is no singularity of the coefficients for either finite or infinite values of s; if the coefficients are uniform, they then can only be constants. The dominant equa- tion is then effectively the same as the original equation ; the investigation is still applicable, but it furnishes less information as to the result than a method which will be indicated later (§ 6). Note 2. The preceding proof is based upon that which is given* by Fuchs in his initial, and now classical, memoir on the theory of linear differential equations. * Creile, t. lkvi (1866), pp. 133—135. y Google 5.] A SYNECTIC INTEGRAL 11 The theorem can also be established by regarding it as a particular case of Cauchy's theorem, which relates to the posses- sion of unique synectic integrals by a system of simultaneous equations. If _ ii"w ™'' ~ ~3^ ' the homogeneous linear equation of order the system -i- = Wj+i, for s = 0, 1, .... (is These equations possess integrals, expressible as regular functions of if—?, such that w„, Wi, ...,w™_, assume arbitrarily assigned values when e — ^, and the integrals are unique when thus determined : which, in effect, is the theorem as to the syuectic integral of the hnear equation*. Note 3. A different method for establishing the existence of the integrals, though if. does not indicate fully the region of (a = ^0, 1 »-I), 1 be replaced by m-^2, -i-pmW< their convergence, can be based upon Giintherf. It consists in the adoption a suggestion of another i made by subsidiary equation where ... +-fm'V, *'-{^-'^r for^=l, ... its integrals are The advantage of this form of equati< explicitly given in the form an is that „.f, -i^fr where a- is a root of the equation (7 (t - 1) ... (<7 -m + 1) = - rjlf,^ (o- - 1) ... ((7 ^ m + 2) + ^M,a- (<7 - 1) . . . (ff - m + 3) + . . . + {- l)™-'r"'-W™_,.7 + (- l)'"r™M^. * See Part ii of this Treatise, gg 4, 10—13. I CreUe, t. csviii (1897J, pp. Sol— 3.13 ; see also some remarks thereupon b; Fuohs, a.,pp. 354, 353. y Google 12 EXAMPLES [5. If a root a is multiple, the corresponding group of integrals is easily obtained*. The construction of the actual proof on the foregoing lines is left aa an exercise. Ex. 1. Consider the equation h arity of the coeffl- 1 d n P y to the immediate po he coefficients of the a, idi unity. The equa- gr wh h ries of powers of z q y d te mm d hy the conditions d & wy constant.^. To which then must be an identity. In order that the coefficient of 2" may vanish after substitution, we must have (« + a)(m + I)6„^.5-{«2 + «-^)6„=0, Now hy the initial conditions, we have h^ = a, 6i = 3; hence " See my Treatine on Differential Equations, %% 47, 43. y Google 5.] EXAMPLES and, similarly, i products baing taken for integer values of s from 1 to m. The syaectic integral satisfying the initial conditions is both series, if infinite, converging for values of i such that | ^ | < 1- The best known instance of this equation is that whict is usually asso- ciated with Legendre's name : k then isp(p + l), ondp (in the simplest form) is a positive integer. If p be an even integer, all the coefBcients b^„, for 2m > p, vanish, so that the quantity multiplying a is then a jwlynomial ; the quantity multiplying j3 is an infinite series. If p be an odd integer, all the coefBcients fiam + u f"r 2ni.+ l >p, vanish, so that the quantity multiplying 3 is then a polynomial ; the quantity multiplying a ia an infinite aeries. In all other cases, the quantitiea multiplyii^ a and (3 are, each of them, infinite aeries ; in every instance, the aeries converge when | z | < 1. £V. 2. Obtain the syncctic integral of the equation (which includes Bessel'a equation as a special case), with the initial conditions that w = a, -T =0 when 2=c, where \o\ > 0. Ex. 3. Determine the synectic integral of the equation of the hyper- geometric series "(l-)^ + {r-(.+»+l).)s-*— », the initial conditions being that w = A, -j- =B, when i=^. E». 4. Determine the synectic integraJa in the domain of £=0, by the equation with the initial conditions (i) that w=l, rT-=0, when z=0 ; (ii) that w = 0, -£-=h wben 2 = 0. £x. 5. Prove that the synectic integral in the domain of 3 = 0, by the equation y Google 14 EQUATIONS WITH [5. with the initial conditions that vi=l, -j- = 0, when ^ = 0, is and if the term in w iuvolving s" he — j s", then c^, = (,»-2 + (2»-^-« + l)»"-Ha3»-«-(»-4)2-=-^™-i| «-'= + .... Prove also that the primitive can he espressed in terms of Bessel's functions of order zero and argument — «="' . &.-. 6. The equation with constant coefficients may he taken in the form which converges everywhere in the finite part of the piano : and «„, ..., (fm-i, are the arbitrarily assigned initial constants. Substituting in the diHerential equation this value of w, and equating coefficients of — 2", wo have The expression of the coefficients a„, Om+i, ■■■ in terms of «(,, «!, ,.., "m-i depends (by the solution of the foregoing difference-equation) upon the algebraical equation "When the roots of ^(e)=0 are different from one another, let them be denoted by a„ a^, —, a^; and in connection with the m arbitrary constants flg, tfi, ..., am_i, determine m new constants ^j, A^, ..., A^, by the relations The determination is unique ; for on solving these m. relations as m, linear equations in A^, ..., A^, the determinant of the right-hand sides is which is equal to the product of the differences of the roots and is therefore not zero. Hence, as the constants a^, c,, .... a„_i ore arbitrary, the m new y Google 5.] CONSTANT COEFFICIENTS 15 constants A^, ..., A^, when iised to replace the former set, can be regarded aa m independent arbitrary constants. With these constants thus determined, = £1 2 on'" + ''-'^K + <'z 2 a^"'*''-^jl^+ ... +c^ S o^'^A^, for all values o! n. When n = 0, we have 2 Ofi"'^^ = CiO„_| + Caa,„,5+ ... +c,„ag = a„^; when n= 1, wo have and so on, the general result being that for all values of n. Hence = 2 (J.a,' + ^s''/+ ■■■ +^^<%*)|^ the customary form of the solution, A^, ..., A^ being m independent arbitrary constants. Ex. 7. Apply the preceding method to obtain a similar expression in finite terms, when the roots of the equation ij>{6)=0 are not all different from one anotiser. 6. A different method of discussing the linear equation with constant coefficients has been given by Hermite. Taking the equation, as before, in the form we associate with it the expression (^) = r" - (Ci?™-^ + c.r™^^ + . . . + c^). Denoting \)y f{K) smy polynomial in l^, let integration being taken round any simple contour in the i^-plane. y Google 16 hermitb's method for equations [6. In the first place, the degree of the polynomial /(5') may be taken to be less than m. If initially it is not so, then we have on division, g (^) being a polynomial, and /, {^) a polynomial of order less than that of 0, that is, less than m. Now |<!<<,(0<i?-o, round any simple contour in the i^-pla-ne ; in the remaining inte- gral, the polynomial is of the form indicated. Accordingly, f(X) will be assumed to be of order less than m. We have taken round the same contour ; because /(f) is a polynomial and the integral is taken round a simple contour in the J;'-plane. Thus TT is a solution of the equation. The only restriction upon f{^) is that, effectively, its degree must be less than m. It may therefore be taken as the most general polynomial of degree m — 1 ; in this form, it will contain m disposable coefficients which can be used to satisfy the initial conditions. Let these conditions require that, when x = 0, the variable w and its first m — \ derivatives acquire values ko, hi, .... km~-i respectively ; then we determine /(i^) as follows. Since we shall dmw the simple contour in the f-plane so as to enclose the origin ; and then the preceding relation shews that, when y Google WITH CONSTANT COEFFICIENTS ^{0 is expanded it. descending powers of ?, the coefficient of ^-r-1 i s kr ; so that, as m. •no it liolds for ■ ^''j + h + r = 0, 1, ... , w i-l. we have and therefore /(?)=•(■«) If +1 + .....|? + . ...}. As /(^) is a polynomial in f, all terms involving negative powers of ^ must disappear, when multiplication is effected on the right- hand side ; and therefore /(O ="S^ k, {^^-' - (C.r"'-'^ + .. . + C^r-,)], the coefficient of k^-i being unity. If therefore w and its first s derivatives are aii to acquire the value zero when z=0, then the degree of the polynomial f{^) is m — s — 2. In order to obtain the customary expression for W, let the contour be chosen so as to include all the zeros of ^(£f). Let a^ be a zero, and let its multiplicity be w,, so that ,(,(f).(!:-.,)".f(r), where the roots of ^i (^) are the other roots of ip (^). Let /«)_ 41. +_^:-_+ I ^y. ,/■«) *«)"?-".(?-«■)■ (f-".)-*.(0' jl'ii, -^'ai, ..., being constants, and /i (0 a polynomial of order m — Ki — 1. So far as the first tii terms are concerned, their contribution to the value of the expression for W is given by taking a contour round a, only. We then have 2lih Jf") -(r-l)!' ^ on changing the constants; and therefore the part, arising through the root Ki of multiplicity n-i, in the expression for the integral is yGoosle 18 hermite's method for equations [6. involving a number of constants equal to the multiplicity of the root. This forra holds for each root in turn ; and therefore the number of constants is the sum of the multiplicities, that is, it is equal to m, the degree of <l> (f ). But m is the number of arbi- trary constants in /(?), when it is initially chosen: these can therefore be replaced by the constants A in the expression S (jIj + ^=2 + . . . + A„£"-") 6"^ the summation extendir denoting the occurs when a another. ■ the roots a of (j){^) = Q, and n lultiplicity of a. The simplest ease, of course, the roots of ^{f) = are different from one The method can be applied to the equation where F(!) ia any function of s. Consider where </)((;■ 1 a tl in f with (u l,.n v. integratioD esteu If <l>(0 = 0. Then -/••'%?« e as b fore / f) 15 a polvnumial i coeftic ent^ f he powers ut f, and ■0 t u tl at icJe-i ill the louts ot ^™,r,/C,J)«-Oi li n auceession, until we have C^J{^,Odi=0; (fo™ /{•,0'K-O- -/'■ ■w*+/^)'""r/"'"«- y Google 6.] WITH CONSTANT COEFFICIENTS 19 Heoce, remembering that /(s, is a polynomial in f and that therefore we hiive W as a solution of the given equation if, in addition to the other cocditioi:iK, which are that forj-=2, 3, ..., m, we have Now as the contour embraj;es all the roots of <}> {0, we have* for r = % .,., m ; ao that, taking where d (s) ia a function of z at ouv disposa], we satisfy the ni — 1 formal conditions unconnected with F(z) ; and then 6 (?) must be such that But as «»-3i-,-f»- /(•,C)-S'»+5i;./'«""^-f(»)'i«, and therefore Hence where j'(f)is, so far as concerns this mode of determining /(e, f), any function of f, and integration with r^ard to u ia along any path that enda in s. When F (e) ia zero, / (s, f ) reducea to ff (f) ; and then the solution of the differential equation shews that ji(f)is a polynomial in £, of degree not higher than to — 1, Aecordii^ly, as ff(^) is independent of i, we take it to be a polynomial of degree n^— 1 in f, with arbitrary conatanta for the coefficients ; and then the integral of the equation has the form y Google 20 CONTINUATION OF THE [6. where the f t t t n est nda nd any mple contour including all the roots of fji (i)=0 Tj d tl teg at n t nds from any arbitrary initial point along J p tl (the pi th b ttcr) to 2. Tho ai 1 nt gral th [ n f Wis clearly the complementary function, and th d ul 1 mte^ri,! h p t ilar integral, in the primitive of the differential equation. The expression can be developed into the customary form, in the same way as in the simpler case when ii* (2) vanishes. Hermite's investigation, based upon Cauchy's treatment by the calculus of residues as espounded in the Exercices de Math^mMiqum, is given in a rnei in Darboux's BvXl. des Sciences Math., 2"" S^r. t in (1879), pp. 311—325 : followed by a brief note {L c, pp. 325 — 328), due to Dai'bous. A mci by Collet, Ann. de I'tc Norm. Sup., 3™ Ser. t. IV (1887), pp. 129—144, may also be consulted. The Process of Continuation applied to the Synectic Integral. 7. The synectic integral P(z— ^) is known at alt points in the domain of J", being uniquely determined by the assigned initial conditions at if. So long as the variable remains within this domain, the integral at z does not depend upon the path of passage from ^ to s, so that the path from f to z can be deformed at will, provided it remains always within the domain. Let ^' be any point in the domain ; then the values of the integral and its first m — 1 derivatives at £" are uniquely determined by the initial conditions at f, and they can themselves be taken as a new set of initial conditions for a new origin ^'. Accordingly, construct the domain of f ' ; and, with the values at f taken as a new set of initial values, form the synectic integral which they determine. As the new initial values are themselves dependent upon the initial values at ^, the synectic integral in the domain of ^' may be denoted by Pi (2 -5^. 0- If the domain of ^' lies entirely within that of ^ (it then will touch the boundary of the domain of ^ internally), the series Pi (e — ^', must give the same value as P (z — ^): for every point z in the domain of if' is then within the domain of f, and it is known that the synectic integral is unique within the original domain. If part of the domain of if' lies without that of ^, then in the remainder (which is common to tho two domains) the series Pi must give the same value as P. But in that part which is outside, the series Pi defines a synectic integral in a region where y Google 7.] SYNECTIC INTEGRAL 21 P does not exist ; it therefore extends our knowledge of the integral, and it is a continuation of the synectic integral out of the original domain. Let Z be any point in the plane; and join Z ko i^ hy any curve, drawn so as not to approach infinitesimally near any of the singularities of the coefficients in the differential equation. Beginning with if, construct the domains of a succession of points along this curve, choosing the points so that each lies in the domain of a preceding point and each new domain includes some portion of the plane not included by any previous domain. Owing to the way in which the curve is drawn, this choice is always possible and, after the construction of a limited number of domains, it will bring Z within a selected region. With each domain we associate its own series ; so that there is a succession of aeries, each contributing a continuation of its predecessor. We can thus obtain at ^ a synectic integral of the equation, which is uniquely determined by the initial values at f and by the path from 5" to Z. Further, taking the values of the integral and its first m — 1 derivatives at ^ as a set of new initial values, and taking the preceding curve reversed as a path from Z to if, we obtain at if the original set of assigned initial values. To establish this state- ment, it is sufficient to choose the succession of points along the curve in the preceding construction, so that the centre of any domain lies within the succeeding domain, and to pass back from centre t<) centre. Stating the proposition briefly, we may say that the reversal of any path restores the initial values. By imagining all possible paths drawn from any initial point if to all possible points z that are not singular, we can construct the whole region of continuity of the integral, as defined by the differential equation and by the initial values arbitrarily assigned at if : moreover, we shall thus have deduced all possible values of the integral at z, as determined by the initial values at if. It is clear, from the construction of the domain of any point and after the establishment of a synectic integral in that domain, which can be continued outside the domain (unless the boundary of the domain is a line of singularity, and this has been assumed not to be the case), that the region of continuity of the integral is bounded by the singularities of the coefficients. As has already y Google 22 DEFORMARLE [7. been remarked, these singularities are called the singulanties of the equation. Thus all the critical points of the integral are fixed points ; and if the equation be taken in the form l-?™w where the functions q^, ..., q^ are holomorphic over the finite part of the plane and have no common factor, these critical points are included among the roots of qo, with possibly z = 'Xi also as a critical point. The value of the integral at an ordinary point near a singularity has been obtained as a synectic function valid over the domain of the point, which excludes the singularity. In later investigations, other expressions for the integral at the point will be determined, when the point belongs to a different domain that includes the singularity. 8. Any path from ^ %o z can be deformed in an unlimited number of ways : and it is not inconceivable that these deforma- tions should lead to an unlimited number of values of the integral at z, as determined by a given set of initial values : but the number is not completely unlimited, because all paths from ^toz lead to the same final value at z with a given set of initial values at £", provided they are deformable into one another without crossing any of the singularities. To prove this, consider a path from f to z, drawn so that no point of it is within an infinitesimal distance of a singularity, and draw a second path between the same two points obtained by an infinitesimal deformation of the first; no point of the second path can therefore be within an infinitesimal distance of a singularity. On the first path, take a succession of points z-i, z^, ..., so that 3i lies within the domains of % and of z^, % within the domains of z^ and ir,, and so on. On the second path, take a similar succession of points a/, si, ..., near Si, i^a, ... respec- tively, in such a way that s/ lies in the part common to the domains of ^ and s,, while z, is in the domain of zl\ Sj' in the part common to the domains of z, and z^, while z^ is in the domain oi si; and so on. Join z-iZ-l, z^^, ... by short arcs in the form of straight lines. Now we have seen that, in any domain, the path from the centre to a point can be deformed without affecting the value of the integral at the point, provided every deformed path lies within y Google 8.] PATHS 23 the domain. Hence in the domain of ^, the path f^i gives at ^, the same integral as the path ^z^'^i. This integral furnishes a set of initial values for the domain of Sj ; and then the path S]Zs gives at % the same integral as the path s-iS^s^z^. Consequently the path fsi^2 gives at z^ the same integral as the path fs/^j, followed by z-tZjZ^Zi. But the effect of z^Zi followed at once by z^z^' is nul, because a reveraed path restores the values at the beginning of the path ; and therefore the path i^z^z^ gives at z^ the same integral as the path t^z^ziz^. And so on, from portion to portion; the last point on the first path is z, which also is the last point on the second path; and tlierefore the path tz,z^...z gives at z the same integral as the path t^(z^...z. Now take any two paths between if and z, such that the closed contour formed by them encloses no singularity of the equation. Either of them can be changed into the other by a succession of infinitesimal deformations : each intermediate path gives at z the same integral as its immediate predecessor: and therefore the initial path and the final path from ^ to s give the same integral at z ; which is the required result. If however two paths between ^ and z are such that the closed contour formed by them encloses a singularity of the equation, then at some stage in the intermediate deformation the curve will pass through the singularity, and we cannot infer the continuation along the curve or the deformation into a consecutive curve as above. It may or may not be the case that the two paths from X,i>a z give at z one and the same integral determined by a given set of initial values ; but we cannot assert that it is the case. Accordingly, we may deform a given path without i the integral at the final point, provided no singularity is c in the process. Moreover, in order to take account of different paths not so deformable into one another, it will be necessary to consider the relation of the singularities to the function represent- ing the integral : this will be effected in a later investigation. When two paths can be deformed into one another, without crossing any singularity, they are called reconcileable ; when they cannot .so be deformed, they are called irreooncileable. If two irreconcileable paths lead at z to different integrals from the same initial values at f, the closed circuit made up of the two paths leads at ^ to a set of values different from the initial values. y Google 24 FORM OF THE [8. These new values can be taken as a new set of initial values : when the same circuit is described, they are not restored, so that either the old initial values or a further set of values will be obtained : and so on, for repeated descriptions of the ciicuit. By this process, we may obtain any number, perhaps even an unlimited number, of sets of values at ^ deduced from a given initial set ; and thus there may be any number, perhaps even an unlimited number, of values of the integral at any point e. Consider any path from f to s ; and without crossing any of the singularities, let it be deformed into loops, drawn from ^ to the singularities and back, (these loops coming in appropriate success- ion), followed by a simple path (say a straight line) from f to s. The final value of the integral at z is determined by the values at f at the begiuning of the straight line, and these values are deducible from the initial values originally assigned. Hence the generality of the integral at z is not affected by taking any particular path from f to z, provided complete generality he reserved for the initial values : and therefore, from this aspect, it will be sufiBcient to discuss the complete system of integrals as arising from com- pletely arbitrary systems of initial values at an ordinaiy point. This investigation relates to properi^ies of the integrals, which will be found useful in discussing the effect of a singularity upon a given integral ; it will accordingly be underiiaken at once. 9. It has already been remarked that the synectic integral, determined by the arbitrary constants which are assigned as the initial values of the function and its derivatives, is linear and homogeneous in those constants: so that, if /a,,, jUia, -.., fiim denote the arbitrary constants, and w-^ denotes the synectic integral which they determine in the domain of au ordinary point i^, we have where ii,, Mj, .... m™ are holomorphic functions of 3 — t^, not involv- ing any of the arbitrary coefiicients /i. Take other m — 1 sets of arbitrary constants fi, such that the determinant I -".I , ^i. Mm , =A(Osay, yGoosle 9.] SYNECTIC INTEGRAL 2-5 is different from zero. Each set of m constants, regarded as a set of initial values, determines a synectic integral, in the domain of ^; as the quantities v^, u^, ..., u^ in the expression for Wi do not involve the arbitrary constants determining w^, it is clear that the expressions for these other m — 1 integrals are Let Msi denote the minor of jist in the non-vanishing determinant A(?); then from the expressions for the m integrals w,, ..., w^ ^"^ terms of Wi, ..., u^, we have ^.{^)u=^MuVh + M^t'a}-,-^--+M^ty'm, (t=l,...,m). Now any other synectic integral, determined in the domain of ^ by assigned initial values $i, B^, ..., 5,„, is given by where the constants & are given by :«,! (r-1, ».). These constants S- cannot all vanish, when the constants ^i, 9^, ..., 6m are not simultaneous zeros : for the determinant of the minors Mri is {A (?)|™~', and therefore is not zero. Accordingly, any integral can be expressed as a linear combination of any m integrals, provided the determinant of the initial values of those m integrals and their first m — 1 derivatives does not vanish. But it is not yet clear that the integrals w-^, ..., w,„ are linearly inde- pendent of one another; until this property is established, we cannot affirm that the expression obtained is the simplest obtain- able. Consider therefore, more generally, the determinant of the m integrals and their first m~l derivatives, not solely at f but for any value of s in the domain of ^, say A{z)^. yGoosle 26 A PECIAL DETERMINANT [9. When 2= f, it becomes the determinant of initial values denoted by a (0- We have d^^^) d'"iu, d™'-'^'W-i dt - d^ • dz'^i *"' d^Wi rf^-^i/l. d!f • d^" "'■ d"*wm i— %« -T^- d»»-- "' =P.aW on substituting tor ^,..., ~^-™ their values in terms of the derivatives of lower orders as given by the equation. Hence Now within the domain of i^, the function jj, is regular, being of the form P,{z — ^); hence the integral in the exponent of e is of the form R{z — t,), where it is a regular function that vanishes when z=^t. Consequently the exponential term on the right-hand side does not vanish at any point in the domain of 5'; also A(f) is not zero ; so that A (z) has no zero within the domain of f. Moreover, each of the quantities w^, ..., w„ is a holomorphic function of z — ^ in that domain, so that A{z) is holomorphic also; hence A(3) has no zero and no infinity within the domain of the ordinary point C i (z) may vanish a any region of Ak a matter of fact, the only points where become infinite are the singularities of pj. For 3 of the functions w,, ..., !(i,„, we have 4j,)_ jji* 4(0 the path from f to s lying within that region, v/hile s is not i the domain off. If a be one of the singularities of pi, the expression of pj i any part of an annular region round a as centre is of the form where the number of terms in according as the singularity ii f s-a is finite or infinite, jntial ; and g (s) is hoio- y Google 9.] FUNDAMENTAL SYSTEMS morphio in the vicinity of a. Taking the simplest c ix^=a^= ... =0; then .'=•*-(!-)>- shewing that a ia a zero of A (2) if the real part of a, be positive, and that it is an infinity of A iz) if the real part of a, be negative. More generally, the nature of A (2) in the vicinity of any singularity a depends upon the character of Pj in that vicinity : in tte case of the above more general form, a is an essential singularity of A (2). Fundamental Systems of Integrals. 10. The linear independence of w,, ..., Wm, and the property that A {£) has a finite non-zero value at any point in the plane ■which is not a singularity of the equation, are involved each in the other. It is easily seen that, if a homogeneous linear relation between Wi, ..., win, of the form CiWi + . . , + c^'.y™ = were to exist, the quantities c,, ..., c^ being constants, then A {z) would vanish lor aii values of s. The inference is at once established by forming the m — 1 derived equations and eliminating the m constants Ci,..., c,„ between the m equa- tions which involve them linearly: the result of the elimination is A(.)-0. Hence if, for any set of integrals Wj,..., w™, the determinant A{3) does not vanish (except possibly at the singularities of the equation), no homogeneous linear relation between the integrals exists. To establish the inference that, if A (a) does vanish for all values of z, a homogeneous linear relation between Wi, ..., w^ exists, we proceed as follows. In the first place, suppose that some minor of a constituent in the first column of A {«), e.g. the minor of —j -J^ in A {z), say y Google 28 A FUNDAMENTAL SYSTEM [10, Ai ie), does not vanish for all ordinary values of 2 ; and take m quantities j/i, ..., ym, the ratios of which are defined by the relations From the hypotheses that A (s) = and that Ai does not vanish, it follows that ' ds""-' Because of the assumption that A^ does not vanish, the ratios 2/™' ym' '"' 2/m are determinate finite functions of s. Differentiate the first of the relations: then, using the second, we have j>i + ...+)/Jw« = 0, where 3// denotes dy-ffdz, for the n values of r. Differentiating the second of the relations, and using the third, we have , dw, , dw,„ and so on, up to obtained by differentiating the last of the postulated relations and by using the deduced relation. We thus have m — 1 relations, homogeneous and linear in the quantities y-^, .--, y^ \ in form, they are precisely the same as the m — 1 relations, which are homogeneous and linear in the quantities j/i, ..., i/,„. Hence, as A, does not vanish, we have *=.&',, (,.= 1,2 m-l). ;©= y Google 10.] AND ITS DETERMINANT 29 SO that — -^constant = ,—''■, (j- = 1, 2, .... m — 1), where Xi, ..., \a~i> ^m ^^^ simultaneous values of y,, ..., ym-\> y-m for any particular value of s : that is, the quantities X are con- stants. This particular value of s is at our disposal; we may assume that X™ is different from zero, because the ratios of ^i , . . . , i/ni_, to ym are determinate and finite. Now hence XlW, + . . . 4- X^Wm — 0, that is, a linear relation exists among the quantities w, if A (z) is zero, and some minor of a constituent in the first column does not vanish. Next, suppose that the minor of every constituent in the first column vanishes : in particular, let Ai {z) = 0, for all ordinary values of z. Then A, (z) is a determinant of m — 1 rows and columns, constructed from m — 1 quantities Wi, ..., w„^^ in the same way as ^{z), a determinant of m rows and columns, is constructed from the m quantities w,, ..., Wm- The preceding analysis shews that, if some minor of a constituent in the first column of Ai (z) does not vanish for all ordinary values of z, then a relation where k, ,...,«ni-i ^^^ constants, is satisfied: so that a linear relation exists among the quantities w, and it happens not to involve ic^- Lot the process of passing from A {z) to A, (s), fi-om A, {z) to a corresponding minor, and so on, be continued; the successive steps are effected by removing the successive columns in A{3) beginning from the left and by removing a corresponding number of rows. At some stage, we must reach some minor which is not zero tor all ordinary values of e : so that y Google 30 THE NUMBER OF [10. vanishes whea s = 0, 1, ..., r, but is different from zero when s = r + l. Then the earlier analysis shews that a linear relation of the form /JlW, + . , . + pm^VJr„^, = exists, where pj,..,,p„^, are constants: in effect, a linear homo- geneous relation among the quantities Wi, ..., Wm which happens not to involve Wm-i+i, ■■-, w™- Hence, if the determinant A(^), constructed from ike m integrals w^, ..., w^, vanishes fiyr all <yrdinary values of z, there is a komoffeneo'us linear relation between these integrals. Integrals are sometimes called independent when they are linearly independent, that is, connected by no homogeneous linear relation ; but the independence is not functional, because all the integi-als are functions of the one variable z. A set of m linearly independent integrals w is called a fundamental system. ; and each integral of the set is called an element or a member of the system. The determinant A (a), constructed out of a set of m integrals, is called the determinant of the system; so that the preceding results may be stated in the form ; — If the determinant of a set ofni integrals vanishes for ordinary (that is, non-singular) values of the variable, the set cannot constitute a fundamental system ; and the determinant of a fundamental system does not vanish for any non-singular value of the variable. 11. We now have the important proposition: — Every integral, which is determined by assigned initial values, can be expressed as a homogeneou^s linear combination of the elements of a fundamental system. Let W denote the integral determined by the assigned values at if, taken to be an ordinary point of al! the coefficients in the differential equation ; and let w, , . . . , Wm he a fundamental system. Let constants c,,...,Cm be deduced such that, when 3 = ^, we have W= 2 CaWa \ y Google 11.] LINEARLY INDEPENDENT INTEGRALS 31 This deduction is uniquely possible; because the determinant of the quantities c on the right-hand aides is the determinant of a fundamental system, and therefore does not vanish when z=^. Thus W ~ i, Ct,WK is an integral of the equation; this integral and its first m — 1 derivatives vanish when 3=^; so that it vanishes everywhere (Cor, I, § 5), and therefore the constants c being properly determined as above. Coe. I. Between any m + 1 branches of the general solution, there must be a homogeneous linear relation. For if m of them be linearly independent, the remaining branch can be regarded as another integral : by the proposition, it is expressible linearly in terms of the other m. Cor. II. Any system of integrals u^, ..., «,„ is fundamental if no relation exists of the form where A^, ..., A^ are constants. For taking a fundamental system W], ..., Wm, we can express each of the solutions u in the form Mr = «lr«'l+ ■■■ + IhnrVm. (r=l, 2, .,., m), where the coefficients a are constants. If G denote the determ- inant of these m^ coefficients, C must be different from zero: for otherwise, on solving the m equations to express Wi in terms of Ml, ,.., Mfli, we should have a relation of the form A^Uj+ ... + .4„M„ = Cwi = ; and no such relation can exist. If, then, A^ (e) denote the deter- minant of the set of integrals u, and if A„ (s) denote that of the fundamental system w^, ..., Wm, we have A.W-CA,W, by the properties of determinants. Now C does not vanish, nor does A,j (s) at any ordinary point in the plane ; hence A„ (2) does not vanish at any ordinary point in the plane, and therefore Ui, ...,u,a ^^e a fundamental system of integrals. y Google 32 A SPECIAL [11. The result may be stated also as follows : If m integrals u be given hy equations u =a,iL,+ +« ' „ <<=1 m) where the deteimmant of the coe_ffnienfs a i/, nd ^eio and t/e integrals w are a fjnda mental system then the bystem of mteQiaU u is also fundamental 12. One paiticuHi fundamental sjstem for the difteiential equation can be jbti.ined as follows Let w be a sppciil mtegial of the equation that is an integral deteimioed 'h^^ an> special set of initial conditions, and substitute w = wjvde in the e(juation : then v is determined by the equation Similarly, let v^ be a special integral of this new equation, with the appropriate conditions ; then substituting V = vjudz, we find that the equation, which determines u, is of the form where m-ldv, '■^ = ^^ ^rf7- And so on. It is manifest that the quantities w,, wjv^dz, w,f(vjihds)ds, ... are integrals of the original equation. Moreover, they constitute a fundamental system ; for, otherwise, they would be linearly connected hy a relation of the form CiW, + dwjvjds + C3W,j{vJu,dz) dz+ ... = 0, that is, , Cj + c.jjv,dz + cj(vju,dz) dz+ ... = 0. y Google 12.] FUNDAMENTAL SYSTEM 33 When this is differentiated, it gives Cj-Ui + Civjujds + ... = 0, that is, c.i + c,Juidz + ... = 0. Effecting m — 1 repetitions of tliis operation of differentiating and removing a non-zero factor, we find as the result at the last stage. Using this in connection with the equation at the last stage but one, we have c™-, = 0. And so on, from the equations at the various stages, we find that all the coefficients c vanish. The homogeneous linear relation therefore does not exist : the system of integrals, obtained in the preceding manner, is a fundamental system. As an immediate corollary from the analysis, we infer that ^i. Vifu,dz,... constitute a fundamental system for the equation in v ; and so for each of the equations in succession. The determinant of this particular fundamental system is simple in expression. Denoting it by A, and denoting by Ai the determinant of the fundamental system of the equation in v, we have, as in § y, ^Pi' IdA i^'dz 1 dAi _ _ m dwi A dz Ai dz ' where Xi is a constant. Similarly, if A3 denote the determinant of the fundamental system of the equation in u, we have y Google 34 FORM OF THE DETERMINANT [12. and so on. The last determinant of all is the actual integral of the last of the equations ; hence i-Cwi™?)j"'-'u,™-^.., where (? is a constant. Moreover, A is the determinant of a par- ticular system, so that C is a determinate constant. It is not difficult to prove that and therefore conaequtntly, Sx. Verify the last result, as to the form of A, in the case of (i) Legendre's equation : (ii) the equation of tte hjpei^eometric series : (iii) Bessel's equation. y Google CHAPTER II. General Form and Properties of Integrals near a Singularity. 13. We have seen that, within the domain of an ordinary point, a synectio integral of a linear differential equation is uniquely determined by a set of assigned initial values ; and that the said integral can be continued beyond that domain, remaining unique for all paths between the initial and the final values of the variahle which are reconciteable with one another. When the variable is permitted to pass out of its initial domain though returning to it for a iinal value, or when two paths between the initial and the final values are not reconcile able, the various propositions that have been established are not necessarily valid under the modified hypothesis : it is therefore desirable to con- sider the influence of irreconcileable paths upon an integral, still more upon a set of fundamental integrals. Remembering that any path is deformable without affecting the integral if, in the deformation, it does not pass over a singularity, we shall manifestly obtain the effect of a singularity, that renders two paths irreconcileable, by making the variable describe a simple circuit, which passes from the point z round the singularity and returns to that point z, and which encloses no other singularity. Let a be the singularity round which the simple closed circuit is completely described by the variable. Let m;,, ..., in^ denote a fundamental system at z ; and suppose that the effect of the 3—2 yGoosle 36 EFFECT OF A SINGULARITY [13. circuit is to change the m integrals into w/, ..., Wm respectively. That the set of m new integrals thus obtained is a fuudamental system can be seen as follows. If it were not a fundamental system, some relation of the form 2 k,w; = would exist, with constant coefficients k, for all values of z in the immediate vicinity. In that case, the quantity S krW,' (which is an integral) is zero everywhere, together with all its derivatives, as it is continued with the variable moving in the ordinary part of the plane. Accordingly, let the integral be continued from z along the closed circuit reversed until it returns to z where, by what has been stated, it is zero. The effect of the reversal is (§ 7) to change w/ into w, : and so the integral after the reversed circuit has been described is % krWr, so that wc should have 2 Kw, = 0, contrary to the fact that v\, .,,, w^ constitute a fundamental system. The initial hypothesis from which this result is deduced is therefore untenable : there is no homogeneous linear relation among the quantities w/, ..., Wm, which therefore form a funda- mental system. Since the system Wi, ..., Wm is fundamental, each of the inte- grals w/, ..., Wm is expressible linearly in terms of the elements of that system ; so that we have equations of the form w/ = a:siiy,+ ...-l-«^w™, (s=l, ...,m), where the coefficients a. are constants. As the system Wg is fundamental, the determinant of these coefficients is different from zero : this being necessary in order to ensure the property that W], ..., Wm are expressible linearly in terms of w/, ..., Wm, a fundamental system. Take any arbitrary linear combination of the system, say where the coefficients p are disposable constants ; and denote this integral by u. When the variable desci'ibes the complete closed y Google 13.1 UPON A FUNDAMENTAL SYSTEM 3Y circuit round the singularity, let w' denote the modified value of u, so that U = pjWi + . . . + pmlOm' = Pi 2 a,rW, . + pml a^rWr- It is conceivable that the coefficients p could be chosen so that the integral reproduces itself except as to a possible constant factor; a relation would then be satisfied, 6 being a constant quantity. This rela- tion, in terms of w,, ..., w™, is Pi 2 a^rWr^ ... + pm 2 a^rWr = 9 (piW, + ... + p™Wm), which, as it involves only the members of a fundamental system linearly, must be an identity: the coefficients of w,, ..., w™ must therefore be equal on the two sides. Hence we have Pittis + p2 (0=2 - ^) 4 ■ + PmCtmi ■ ■ + pm«mi If, therefore, $ be determined as a root of the equation the preceding relations then lead to values for the ratios of the constants p for each such root. It is to be noted that, in this equation, the term, which is independent of $, does not vanish, for it is the determinant of the coefficients a ; hence the equation has no zero root. As the equation definitely possesses roots 9, it follows that integrals exist which, after a description of the simple contour round a, reproduce themselves save as to a constant factor. If it should happen that the constant factor is unity, then the effect of y Google 38 THE FUNDAMENTAL EQUATION [13. description of the contour upon the integral is merely to leave it unchaiiged : in other words, such an integral is uniform in the vicinity of the singularity. Propeuties of the Fundamental Equation. 14. The special significance of the equation, in relation to the singularity a, lies in the proposition that the coefficients of ike various powers of 6 in A =0 are independent of the fundamental system- initially chosen for discussion. To prove* the statement, it will be sufficient to shew that the same equation is obtained when another fundamental system is initially chosen. For this purpose, let y,, ,,.,ym denote some other fundamental system; and suppose that, by the simple closed contour round a described by the variable, the members of the system become j^/, ,.,, ym respectively. Then, as both these systems are fundamental, there are relations of the form y/ = 0s,yi + -- + 0^y^, (s = i, ....m), where the determinant of the coefficients /3 is not zero. The equation B = 0, corresponding to j1 = for the determination of the factor 8, is formed from the coefKcients ^ in the same way as A from the coefficients a, so that the expression for B is B = Because each of the sets w„ ,.., w^; i/,, ,.,, i/^; is a funda- mental system, the members are connected by relations of the form y., = y„w, + y^w^+... -1-7™^^, (r^l, ..., m), where the determinant of the coefficients, which may be denoted by r, is different from zero. The quantity is zero everywhere in the vicinity of z ; and it is an integral, which accordingly is zero everywhere in its continuations over the * The proof adopted is due to Hamburger. Crelk, t.'Lxxvi (1873), pp. 113—125. y Google 14.] IS INVARIANTIVE 39 ordinary part of the plane. When it is continued along the simple contour round a, the variable returning to z, the integral is zero there ; that is, Hence and therefore S 2 /SrsJeiWi =22 7rs«siW[. This relation involves only the members of a fundamental system hnearly ; hence it must be an identity. We therefore have 2 ^r67B(= 2 7rsas* say, the relation among the constants holding for all values of r and t. Now forming the product of the determinants P and A, we have 7n. Tis' 7i3. 721. 722. 7S3' 7si, 7aa, 7as. 5ii-7u^. Sk-7iA ... 821 — 7h^, S22 — 7,2^, ■ ■ ■ say ; and similarly, forming the product of B and V, we have ' — 6, ^,2 , /3i3 , .■■ 7ii, 7si, 731. , ji^-0, /3a , ... 7i3. 7^. 732. , ^^ ,0x1 — 0. ■■■ 7i3> 723. 73s. -7n^. S,.,-7,A ... -72A ^^-y^0, ... = -0, yGoosle 40 INVAEIANTIVE PROPERTIES [li. identically. Also F does not vanish ; hence for all values of B. Accordingly, the equation A = is invariantive for all funda- mentaJ systems in regard to the effect of the singularity a upon the memhers of the system: it is called* the fundamental equation belonging to the singularity a. We note that its degree is equal to the order of the differentia! equation. While the equation is thus invariantive for all fundamental systems, the actual invarianee of one of its coefficients is put in evidence, either when the differential equation of § 2 is initially devoid of the term involving -j-;^ , or after the equation has been transformed by the relation SO as to be devoid of the term involving , ^_^ . In A =0,the term, which is independent of 6 is equal to unity, a property first noted by Poincar^f. For when ^i is zero, the determinant A of the fundamental system is a constant, for (§ 9) its derivative vanishes ; it therefore is unchanged when the variable describes a simple closed circuit round the singularity. The effect of such a circuit upon A is to multiply it by the term in A which is independent of 8 : accordingly, that term is unity. The linear equation can always be modified so that the term involving the derivative of the dependent variable next to the highest is absent; and the necessary linear modification of the dependent variable leaves the independent variable unaltered. This change does not influence the law giving the effect, upon the integrals, of a description of a loop round the singularity; and the fundamental equation is independent of the choice of the fundamental system. Accordingly, the coefficients of the various powers of 9 (except the highest, which has a coefficient (—1)™, and the lowest, which has a coefficient unity) are fre- quently called the invariants of the singularity : they are m — 1 in number. • Sometimes also tha charaeteristic equation. t Acta Math., t. iv (1884), p. 202. y Google 15.] OF THE FUNDAMENTAL EQUATION 41 15. There is a further important invariantive property of the determinants A(d), B{ff), viz.: If all minors of order n (and therefore all minors of lower order) in A (8) vanish for a particular value of 6, but not all those of order n + 1, then all minors of order n in, B {&) also vanish for thai, value of 8, hut not all those of order n + \. A minor of order n is obtained by sappresaing n rows and n columns ; accordingly, the number of them is say. Let them be denoted by ftfj, hij, c^, dij when formed from A {0), B (0), r, J) respectively, where * and j have the values 1, ..., /i, these numbers corresponding to the various suppressions of the rows and the columns. Then, regarding D as the product of A and V, we have* dij = Ci,aj, + Gjaaj-i + . . , + Ci^aj> ; and regarding D as the product of B and T, we have dij = biiCj, + bi^Cji+ ... +i'i„Cj„. All the quantities (ifj are supposed to vanish for a particular value of d ; hence for that value all the quantities dij vanish. Assigning to j all the values 1, ..., ^ in turn, we therefore have = Cii&ii +Cis&i3 + ... +c,^bi^ \ =- cAi + Cs^bi2 + ... +c.^bi^ I , = C^ibij + C^s6;s + . . . + c^^bi^ I The determinant of the coefficients of bi,, hi,, ... , V '^ equal tof T\ where x = _C^-i)'_. {m-n-\)\ nV that is, the determinant does not vanish. Accordingly, we must have ht^^O, b^ = (},...,hi^ = 0; as this holds for all values of i, it follows that all the minors ot B{9) of order n vanish for the particular value of 0. * Scott's Determinants, p. 53. + ih., p. lil. y Google 42 ELEMENl'ARY [15. The minors of B (S), which are of order li + 1, cannot all vanish for the value of 0; for then, by applying the result just obtained, all those of A (0), which are of order n+1, would vanish, contrary to hypothesis. 16. A more general inference can bo made. Leaving arbi- trary and not restricting it to be a root of the fundamental equation, the two expressions for dij give holding for all values of i andj. Taking this equation for any one value of _y and for all the /t values of i, we have ^ equations in all, expressing a,-,, a^, ..., iij> linearly in terms of bpq. The determi- nant of coefficients on the left-hand side is T*, as before, and does not vanish ; so that each of the quantities Oj^ is expressible linearly in terms of the quantities ftp,, the coefficients involving only the constituents of V. Similarly, taking the equation ibr any one value of i and for all the ^ values of j, we find that each of the quantities bp^ is expressible linearly in terms of the quantities aj>, the coefficients involving only the constituents of F. If therefore all the quantities a^r have a common factor — 0i, and if that factor be of multiplicity o-, then all the quantities bpq also have that factor common and of the same multiplicity a- ; and conversely. These results associate themselves at once with Weierstrass's theory of elementary divisors*. If {0 — Oi)" is the highest power of 8 — 0-, in A (6), if (6 — O^y^ is the highest power of that quantity common to all its minors of the first order, if (0 — 01)"' is the highest power common to all its minors of the second order, and so on, then (as will be proved immediately) a >a;><7^>...; and («-»,)—', (9- e,)-.--., ... are called elementary divisors of the determinant A (0). It follows from the preceding investigation that the elementary divisors of the fundamental equation, are invariantive, as well as the equation * Berl. Monatsber., (1868), pp. 310—388; Oes. Werke, t. ir, pp. 19—44. See aUo & memoir by Sauvage, Ann. lie l'£c. Norm., 3" S&-., t; vm (1891), pp. 26a— SiO ; and a treatise by Math, EtemenlaTtlieiUr, (Leipzig, 1899). y Google 16.] DIVISORS 4-3 itself; for they are independent of the particular choice of a fundamental system. If the earliest set of minors of the same order that do not all vanish when = Si is of order p, so that they are of degree m.~ p in the coefficients in A, then the elementary divisors are being p in number : and then p is one of the invariantive numbers associated with the particular singularity of the equation. As two of the properties of the invariantive equation, associated with the elementary divisorB, are required, they will he proved here : for full discusaion of other properties, reference may bo made to the authorities quoted. It is easy to obtain the result cr>o-i>irj> ..,, just stated above. For 9^ Z A M = -.!/- wlieie -J r 1^ the muioi ot a^, 6 In A„ there is a factor (0-0,)"', for each of tlie qmiititie-i ■!„ is i fir«t minor ; therefore that factor occurs in their sum and, owm^ to the uimbination of terms, it may have an even higher indes than o-, On the left, the factor in ^ - ^j has the index o- - 1 i hence that is, Similarly for the other inequalities. Again, we know • that any minor of degree p which, can be formed out of the first minors of A {&) is equal to the product of ^p^' {6) by the comple- mentary of the corresponding minor of A {6). Hence, taking p = 2, we have relations of the form A^B2-A^B^ = AC, ora of the first order, and C is a minor of the r of the second order which is divisible by no higher power of 6— 6, than (fl— fl,)"'; the left-hand side is certainly divisible by {6 - S^f"', and it may be divisible by a higher power if the terms combine ; hence that is, Similarly, we have the other inequalities of the set cr-o-^>T^-ir^5:a2-irs>...><ri,_i, so that the indices of the elementary divisors, as arranged above, form a series of decreasing numbers. y Google BRANCHES OF AN ALGEBRAIC FUNCTION" [17. Association of Differentiai- Equations with Algebraic Functions. 17. Before considering the roots of the fundamental equation, it is worth while establishing a converse* of the propositions in § 13, as follows: Let j/i, ,.., ym he m linearly independent functions of s, which are uniform over any simply-connected area not including any critical point of the functions : let the critical points be isolated and let each of them be such that, when a simple contour enclosing it is described, the values of the Junctions at the completion of the contour are given by relations of the form yr'=a«,?i + --. + «™2/m, ('• = 1, ...,m), where the determinant of the coefficients a. is not zero, and the constants may cJiange from one critical point to another : then, the m functions are a fundamental system of integrals for a linear differential equation of order m unth uniform coefficients. It is clear that, if the functions are integraSs of such an equation, they form a fundamental system because they are linearly indepen- dent. On account of this linear independence^ the determinant dz'"^' ' d^"^" d^^' ' dz"'-' ' ■■■■ does not vanish for all values of z. Let As denote the determinant which is derived fi-om A by changing the stii column into -j—^ , '"' rir^ ' ^^^ consider the quantity For any contour that encloses no critical point, A and A, are uniform, so that ps is uniform for such a contour. For a simple contour, which encloses the critical point a and no other, the ' It is given by Tannery, Ann. de VKc. Norm.. Ser. 2"", t. iv (1875), p. 130. y Google 17.] AS A FUNDAMENTAL SYSTEM 45 determinant A after a single description acquires a constant factor R, where R is the (non-zero) determinant of the coefficients in the set of relations ^/r' = «rii/i+--- + a™3/m, (r = 1, ...,m). The determinant Aj acquires the same factor R, in the same circumstances ; and therefore ps is unchanged in value by a description of the contour, that ia, it is uniform for such a contour. As this holds for each contour, it follows that pg is uniform over the plane. The m quantities y,, ..., y^ evidently are special integrals of the equation which is linear and the coefficients in which have been proved uniform functions of s. Corollary. If all the critical points of the functions are of an algebraic character, that is, of the same nature as the critical points of a function defined by an algebraic equation, and are limited in number, then the uniform coefficients p m the differential equation are rational functions of z. For as p^ is uniform, the critical point a is either an infinity, or an ordinary value (including zero). If it is an infinity, it can be only of finite multiplicity; for the critical point is one, where A and A, can vanish only to finite order because of the hypothesis as to the nature of the critical point: that is, the point is then a pole of finite order. Likewise, if it is a zero, the multiplicity of the zero is finite. This holds at each of the critical points of the functions y\,..-,ym\ and the number of such points is finite. Moreover, every point that is ordinary for each of the functions is ordinary for A and Ag and, in particular, A cannot vanish there : so that no such point can be a pole of any of the coefficients p. It therefore follows* that each of these coefficients is a rational meromorphic function oiz. The converse of the corollary is not necessarily (nor even generally) true : it raises the question as to the tests sufficient and necessary to secure that the integrals of a linear equation with rational coefScients should be algebraic functions of the variable. This discussion must be deferred. * T. F., S 43. y Google ALGEBRAIC FUNCTIONS [ir. 46 Ex. 1. The most conspicuous instance arises when the dependent vari- able w is an algebraic function of z, defined bj an algebraic ec[iiation of degi'ee m io w. Ba«h branch of the function so defined is uniform in the vicinity of an ordinary point ; in the vicinity of a branch-point, the branches divide themselves into groups ; and any linear combination of them is subject to the foregoing laws of change (which take a particularly simple form in this ca.se) when z describes a circuit round a branch-point. To obtain the homogeneous linear equation of order m which is satisfied by every root of j''=0, we can proceed as follows. Let ^(s) = be the eliminant of /=0 and =~=0; so that* all the branch-points of the alge- braic function are included among the roots of 0=0, though not every root is a branch-point. By a result f in the theory of elimination, we know that the resultant of two quantics u and v of degi'ee ni. and n respectively in a variable to be eliminated is of the form where Mj and % are of degrees m- 1, «— I respectively in that variable ; and therefore where i/ is of degree m— 2 io w and V is of degree ni- I in n; But / is ? equal to zero for all the values of w considered ; hence .ally derivable from Sjl of the ehminant, ( To the last column, add the first column multiplied by .r™"'^"^, the second multiplied by iii'"+"-2, and so on : a change which does not affect the value of E. The couBtituants in the new last column ace x'^hi, x'^ht, .... xu, K, x^^-^i), i™-2i), .,., XV, v; eipanding E by taking every term in this last column with its minor, eoUeo ting all the terms involving ti into one set and those involving v into naother, we have where v, s of degree m -1 ii y Google 17.] AND DIFFERENTIAL EQUATIONS By m.» i of/=0 which ia of degree m in w, we can , reduce v'f- contains i 10 power of w h^her than the (i m-I)th, say where P, is a polynomial in w of degree not higher than m-1. (If the highest term in / has unity for its coefficient, then P, ia a polynomial in s also.) Aj;ain, d^ Pi dp, 1 dP, F, a^ d^a 02(3) ^ -+-0(2) g^ 02(5) a^ on reduoing to a common denominator ; by u Pj can he made of degree not higher than nj- uniform functions of z. And so on, up to where I'^ is a polynomial in i being uniform functions of e. ns of /=0, the polynomial 1 M, and its coefficients aro 7 of degree not higher than m - 1, the coofflcionta We thus have Among these m equations t m- 1 quantities u^", a)*, ..., iP" the form can, by a linear combination, eliminate the ■^ irom the left-hand aides ; and the result has &•«=' ™^2m 9" where Q„, <2i, ..., §m are uniform functions of s, Thia is satisfied for every root JO of the algebraic equation : and it is of order m. Corotlary. There ia one special case, when the differential equation is of order m - 1, viK., when the algebraic equation ia /=w"'-l-a2)«™-=+ ... +0^=0, BO that the term in w™~' is absent. We then have y Google 48 EXAMPLES OF [17. SO that one of tho in, branches w can he espreaaed ]inearly in terms of the others ; Tannery's r^ult ahewa that the differential equation is theii, of order •not higher than m — 1. In that case, it would be sufficient to take only the m — 1 equations ^-1"*'. (- •»-')• For instance, consider the algebraic equation -u^ + to^M, where u is any function of 2 ; it is to be expected that the liaear differential equation satisfied by each of the three branches of the function defined by this cubic equation will be of the second order, say where A and B are functions of :. We have '-'+')^+*'(J)"-i»- so, substituting in (».+i)5' + JK+i)f+^(«'+«)-o, and using M'^ + 3;(i=ii, wo have Multiplying the right-hand side by {«j* + 1)^, and the left-hand side by its equivalent l + wu — vfi, we have on reduction by the original algebraic equation. This will hold for each of the three roots of that equation, if g«'^=«(iJ«.'+K)+^(-«^-8)l O^lA'd + 1%" + 'ABu y These conditions give the values of A and B ; and the equation for w is easily found to be dho ( uu- u"\d-w_ u'^ "^2"'"UH4 u-)dz *ii«+4'^ where u' and u" are the first and the second derivatives of u. The equation oi the seco 1 de as d ted A 1 When the Igeb a equat on of degree m is of quite g ner 1 fo m the 1 near 1 fierent al equat n sat sfied by t roots is of order m B t when the ^gebra c eq at on has orj sj al form-i, though still rre I ble the d fferent al eq at on in y be of order 1 s th m ; for the y Google 17.] DIFFERENTIAL RESOLVENTS 49 elimination of various powers of w may not require derivatives up to that of order m. The most conspicuously simple ease is that in which the alge- braic equation is where K is a rational function of z ; the differentia! equation is only of the first order. Other cases occur hereafter, in Chapter v, where quantities connected with the roots of algebraic equations of degree higher than two satisfy linear differential equations of the second, order. ^ote 2. The differential equations considered have, in each case, been homogeceoua. If we admit non-homogeneous linear differential equations, viz. those which bave a term independent of w and its derivatives, then in the general case, where /(w, i) has a term in (t"""*, the differential eqviOtion is of order m—1 ordy. This can be seen at once from the elimination of w^, -ui*, ..., a;™-! between ^ ds ^ leading to a (non -homogeneous) linear equation of order m- 1. This result appeal's to have been iirst stated by Cockle*; it is the initial result in the formal theory of differential resolventst. &«. 2. Shew that, when the algebraic equation is the two linear differential equations, homogeneous and non-homogoneous respectively, are ^ _ 3-t-23*i rf«j 3+22= d^ Z-^r^ dz 1'+^'^ ' dm H-2s^ # di s+s^ ^'"' 1+^' E.V. 3. Obtain the differential equations satisfied by efich root of (i) ffj3_3^2^^ = 0; (ii) i!^-Zzw^si=0. Ex. 4. Shew that any root of the equation r-ny = {n-l)x: * FUl. Mag., t. sxi (1361), pp. 37fl— 383. t Foe cefarences, see a paper by Harley, Manch. Lit. and Phil. Memoirs, t. v y Google 50 SIMPLE ROOTS OF [17. (n being greater than 2) satisfies tlie eqiiatiuii wheiw a = l — . What ia the form for « = 2? (Heymann.) Ek. 5. Shew that any root of the equation (« lieing greater than 2) satisfies the equation where the constants a^ arise as the coeffioients in the algebraic equation _s(-iri jA'- when the roots a ro x=(i--,)^^'L^ fori = l, . .., B-. L, and «..- -1=1- Ex.e. then Prove that, if r--V+5y- -ix i^y 2,-g-l rT=0; and explain the decrease in the order of the differential equation. (Math. Trip., Part ii, 1900.) ^ii^■1)amental system of integrals associated with a Fundamental Equation. 18. We now proceed to the consideration of the fundamental equation A = Q appertaining to the singularity a. The simplest ease is that in which the m roots of that equation are distinct from one another, say 6^, 9^, ..., 6^- Not all the minors of the first order vanish for any one of the roots : if they did vanish, the root would be multiple for the original equation. Hence each root 6r determines ratios of coefficients c^, c,.j, ..., Crm uniquely, such that an integral of the equation exists, having the value and possessing the property that y Google 18.] THE FU^DAMENTAL EQUATION 51 where m/ is the value of Ur after z has described a complete simple contour round a. We thus obtain a set of m integrals. These m integrals constitute a fundamental system : otherwise a permanent relation of the form KlMl + Kstf, + . . . + K would exist. This quantity Sk^w^ is = integral ; as it is zero and all its derivatives are zero at and near z, it is zero everywhere when continiied over the regulaj- part of the plane. Accordingly, let z describe a simple closed contour round a: when it has returned to its initial position, the zero-integral is Sk^m/, that is, kA-Hi + icAui +... + K^O,^iim = 0. Similarly, after a second description of the simple closed contour, we have KA'Ur + 'cA^ti, + ... + >c„e„>,^ = 0. made in this way : we Let m — 1 descriptions of the contour 1 (i + K for r = 0, 1, ... zero, we have i all the coefficients / that is, the product of the differences of the roots is zero. This is impossible when the roots are distinct from one another; hence the coefficients «,, ..., k^ vanish, and there is no homogeneous linear relation among the integrals Wi, ..., «,„, which accordingly constitute a fundamental system. The general functional character of these integrals is easily found. Let so that T^ is a new constant, which is determinate save as to any additive integer; as the roots 8i, ..., dm are unequal, no two of the m constants r^, ..., r^ can differ by an integer. Now the quantity y Google 52 EFFECT OF A [18. acquires a factor e "'''', that is, 6^, when z describes the simple complote circuit round a. Hence the quantity returns to its initial value after the variable has described the simple complete circuit round a; and therefore it is a uniform function of s in the immediate vicinity of a, say tf^, so that As this holds for each of the integers i^, it follows that we have a system nf fundamental integrals in the form where tj>,, 02, ..., (f>m o-i'^ uniform functions of z in, the vicinity of a, the quantities r^ are given by the relations j-„=^log^„, and the roots 0,, ..., dm of the fandamental equation are supposed distinct from one another, no one of them being zero. As regards this result, it must be noted that the functions ^ are merely uniform in the vicinity of a : they are not necessarily holomorphic there. Each such function can be expressed in the form of a series of positive and negative pov^ere of z — a, converg- ing in an annular space bounded by two circles having a for a common centre and enclosing no other singularity of the equation. There may he no negative powers of ^ — a, in which case the function >}> is holomorphic at a ; or there may be a limited number of negative powers, in which case a is a pole of <^ ; or there may be an unlimited number of negative powers, in which case a is an essential singularity. Moreover, r^ is only determinate save as to additive integers : it will, where possible (that is, when a is not an essential singularity), be rendered determinate hereafter ; so that, in the meanwhile, the result obtained is chiefly important as indicating the precise kind of multiform charactei' possessed by the integrals near a singularity. 19. Now consider the case in which the fundamental equation A = appertaining to the singularity a has repeated roots, say Xi roots equal to 0,, X^ roots equal to 8^, and so ou^ where ^,, 6^, ... are unequal quantities, and Xj + Xj + . . . = m. It will appear that y Google 19.] MULTIPLK ROOT 53 a gi-oup of linearly independent integrals is associated with each such root, the number in the group being equal to the multiplicity of the root ; that each such group can be arranged in a number of sub-groups, the extent and the number of which are determined by the elementary divisors connected with the root ; and that the aggregate of the various groups of integrals, associated with the respective roots of the fundamental equation, constitutes a funda- mental system. Group of Integrals associated with a Multiple Root oe THE Fundamental Equation. Let K denote any such root of multiplicity a; and let the elementary divisors of A (8) in its determinantal form be {«-«)'—, (e-«)"-'-. .... («-«)'--"•-, («-«)'-■; then the minors of order r (and consequently of degree to — t in the coefficients of ^) are the earliest in increasing order which do not all vanish when d — k. Consequently, in the set of equations T of them are linearly dependent upon the rest; hence taking m — r which are independent, we can express m — t of the con- stants p linearly in terms of the other t, which thus remain arbitrary. Let the latter be pi, ..., p^; then the integral, given by U = p,W, +p2tVi +--. + pm1«m. becomes u = p,W, + p^W,+ ...+p,W„ where Tfi = Wi-l-A:,+i,|W^+i + ... + k„,iW,^ H^2 = Ws + /;,+,, aWr+i + . ■ . + h>i,2^n W, = iVr + &,+,,TWr+. + ■ ■ ■ + ^m.rW™ and the determinate constants k are given by Pr+i = h+,^,p, + k,+,^^p^ + ... + k^+j^. pr-n = h+ii,ipi + kri-i,sP^+ ■•■ +k,+2, Pm =/l^™,iPi +^m,ap2 + ■-- +4»,ti y Google 54 ELEMENTARY [19. being the expressions for the m — r quantities p in terms of the T quantities p which remain arbitrary. Evidently each of the quantities W is an integral of the equation : and they have thc! property w;=kW,. for r = l, ,,., T. Moreover, they are linearly independent; any non- evanescent i-eiation of the form would lead to a relation between w,, ..,,Wm which would be homo- geneous, linear, and non-evanescent, a possibility excluded by the fact that Wi, ,.., if™ constitute a fiindamental system. The only case, in which t = o-, occurs when the indices cr — o-, , CTi — (Tj, ..., tTr-i of the elementary divisors are each unity. In that case, we have obtained a set of integrals, in number equal to the multiplicity of the root. 20, We shall therefore assume that t < cr ; and we then use the integrals W,, ..., W, to modify the original fundamental system w,, ..., w^, substituting them for Wi, ..., w^. When the variable z describes a simple closed contour round a, the effect upon the elements of the modified system is to change them into IT/, W^', ..., W,', «/',+„ ..., w'n, where W,f^ cW^, ■...+8.,«^ for r — 1, ..., T, and s = t+ 1, ..., m. The fundamental equation derived from this system for the singnlaiity a is .4(n) = o, where K-a, ,. .. ,.,. , «-n,. . .... 0.0,. ., K-a .... At.... A„.... . A+.„ fi,^ .„+. -n, A+,„,„... I3,„,,. /3„ , /3„ ,. ., /3.„ A. .« , /3»,,+, ,.., &„-f .(«-n)'A,<ii), y Google 20.] where As « is a root of A (U) of multiplicity <t, it is root of Ai (Xl) of multiplicity er — T ; and a question arises as to the elementary divisors of A^ (fi) associated with k. The elementary divisors of A-^ (fl), which are powers of k — Xi, are (n-«)'-.-", (n-«r-'.-i, (n -«)•—.->, ... being, in each instance, of index less by unity than those of A (H). This result, which is due to Casorati*, follows from the property that .4,(0) is divisible by (li — «)"-'; its iirst minora are divisible by (ii — «)">"''-" and not simultaneously by any higher power; its second minors are divisible by (fi — k)"'"''"^' and not simultaneously by any higher power ; and so on. Thi« property, that all the minors of A^ (Si) of order fi ai'e fliviaible by (jL. 11) '' ' ' and not isimultancously by any higher power, can be proved a follow t Aiy mmoi of order ^ of ^ (Q) must contain at least ra—r-ii of the last m T columns let it contain m-r—ix+a of these columns, where a can range from to /i. It then miiat contain r-o of the first r coJumne. ^im larly it must contain at least m-r— it of the last nt — r rows : let it conta n r /i + a' of these rows, where a' can range from to ^. It then must cu tai r — q' of the first t rows. The minor may be identically zero : if not then ow ng to the early columns and early rows that are retained, it is livisible by (k 2^"", and possibly by a higher power of k — a. Conse- quentlj ome imong these minors are espreasible as the product of {« -fi)'"'' by a Imear combination of minora of Ai (Q) which are of order /i ; the coefii- eienta in the combination are composed of the constants, which occur in the first T — ^ columns and the last m-r rows, and thus are independent of O. But a minor of order it ol A (Q) is not necessarily divisible by a powei' of K - fi with an index higher than <t ; thus (i!-fi)V. polynomial in fi=(s-fi)'""'''.sum of minors of ^, (n). It therefore follows that the power of « - Si common to all those minors of A, {a) is of indei not higher than o- -(t — ^^). ' Comptes Reitdus, t. xcii (I88I). p. 177. + Heffter, MMeitmm in die Theorie der Uitearen DiJfsrenUulyleichvngen. pp. 350—256. y Google 56 ELEMENTARY DIVISORS OF THE [20. Kext, we know that thore axo some minora of the original A (ii) of order t, which do not vanish when JJ = k and which therefore are not divisible by « — Q. Clearly they cannot contain any of the first t rows in A (Si) ; and thus they miat ho composed of sets of to — t columnB selected among the last m - r vows. Take the minors of order )i of any one of these non-vanishing determinants, their number being N^, where = ^; and denote these minora by ifi (/ A=l, ...,N), tl togors /adiurrei Iftttl 11 terat on of i et of ^ columns a d a set f /i rowa o t of the non an sh j, deter ant of order m - t. Let ft Ije the comj leme tarj of Y,^ n ta w determ na t Non take tl e nunor of 4 (Q) nh ch ire of o de /t the number ia iV', a d thej n aj be denoted by o j f o ^ = 1 \ w tl tl e o significance in the integers as for M/j^. Construct an expression say, where J^ is a determinant of order m-r. Then either (i), J^ vanishes identically, owing to identities of rows or columns ; or (ii), J^ ia equal to + A^(il) and therefore is divisible by (^-fl)"'"'', that is, certainly divisible by («-n)°"^ "('■-''*, for (§16) wo have T — (T'l > (T^ — o-^ ^ . . . ^ u^ _ 2 5^ 1 ; or (iii) /ft, when bordered by r — fi of the first rows, and tho first columns in A (11), is a minor of order ^ oi A (JJ) and ia therefore divisible by (k— !J)''i', so that the equivalence of tho two espressiona for the minor of A (Q) gives (k — il)'^f . polynomial in a = (K- ilY~''.Jh, and therefore J^ is divisible by (jt-n)V~t''~'''. Tt thus follows that J,, is divisible by (K-fl)'^» "*''"'*', in every case when it is not aero: and this holds for all values of /(. Taking then rai,aii + n(jja(g + ... + mjwai,v=/i, for /i=i, ..., A' and for one particular value of i, we have a series of A^ linear eijuations in the quantities a;,, ..., 0;^-. The determinant of their coefficients is a power of the non- vanishing detemiinant of order to-t, for it ia a determinant of all its minors of one order : and therefore it doea not vanish. Hence, so far ispoRe'a t k Jl are conce ed e h of the mnoB a, a v is a linea co h n t of / ■/> <kll of 1 e e are d s ble by (k-Q)"«" '' d d he ef -e h of the m o ■s a. a jf s «rta nly divisible by h t power The esult 1 olds lo ea<;l t the val es of It has eea oe t t e i we o k a om no to 11 tl ese n ors of A^ (Q), has a n 1 ot greater ha a /i omb n^ tl e rea Its, ve infer that the highest power of k - Q, common to all the minors of A^ l_il) of order /i, has its index equal to ir - (r - p). y Google 21.] FUNDAMENTAL EQUATION 57 21, The indices of the elementary divisors of A^ {11) at-e ff-o-,-1, <r,-o-2-l, a,-(T,~l, ...; let there be t' of them, where t'$ t, so that the last t — t' of the indices of those of A (il) are equal to unity, on account of the property o- - 0-, ^ 0-1 - o-s ^ o-a - o-a > - . . 3^ T,-, > 1. Then the minors of .^i of order t' (and consequently of degree TO — T — t' in the coefficients of Ai) are the earliest in successively increasing order, which do not all vanish when il = k; conse- quently, in the set of equations Pl'0r.r+, + P-l^r,,^^ + ■ ■■ + pV, /3,,„, = Kp/, (r = r + 1, . , ., to), t' ol' them are linearly dependent upon the rest. Hence taking in — T — t' of the equations which are independent, we can express m — T — T of the constants p' in terms of the other t', which thus remain arbitrary and which may be taken to be p,', . . ., p'^. Now take an integral v = pM+. + ... + p\n-.w..> and substitute for the various coefficients p' in terms of p,', ..., pV- The integral becomes V = p! TF"„ + p/ ^fj^ + - . . + p',- t^lr' ■ where, writing \ — t + t\ we have Wir = W^+r + k+i,rVlk+i + . . - + Un,rWm, forr=l, ...,t'; and the determinate constants i are given by Pr'+s = /j,+B,ip,'+ ... + ^J.+s, r'pV, for 8 = 1, .... m — X, being the expressions of the constants p' in terms of ,0,', .... p',-. Clearly ea<;h of the quantities W-a, ^a, ■■■, W-^' is an integral of the equation. Moreover, they are linearly independent of one another and of W], ..., W,\ for any non-evanescent linear relation of the form F,W, + ... + F,W,^F:W,, + ... + fVTf.y = would lead, after substitution for W-^. ..., Wj, W"„, ..., Wi,- in terms of the original fundamental system Wj, ...,Wm, to a non- evanescont homogeneous linear relation among the members of that system — a possibility that is excluded. y Google 58 SUB-GROUl'S OF [21. As regards the effect, which is caused upon each of these newly obtained integrals by the description of a simple contour round the singularity, we have Tf,,' = W\^, + k^,.rVj\+, + ... + ln,,-y>«' = lcW,r+ V,., where V,. denotes a homogeneous linear combination of TTi, ..., W,. Now no one of the quantities Vy can be evanescent, nor can any linear combination of the form 7iFi + ... +7t'F,' be evanescent : for in the former case, we should have and in the latter (7, w„. + . . . + 7,' w„.y = « {-y, If „ + , . . + y,, r,.o- As Wir and y,W„ + ... +'y.,'W,r- in the respective cases are linearly independent of W^, ..., W,, we should thus have a new integral of the same type as Wi, ..., Wr', and then, instead of having some of the minors of order t in A (H) different from zero when H — k, all of them of that order would be zero, and we should only be able to declare that some of order t + 1 are different from zero : in other words, the number of elementary divisors of A (H) would be T + 1 instead of t. The quantities V^, ..., V^ are thus linearly equivalent to t' of the quantities Wj, ..., W,, say to Wi, ..., TT^'; hence constructing the linear combinations of V,, ..., V^i which are equal to TT,, ,.., Wj' respectively, and denoting by w,,, ..., Wir' the linear combinations of W^, .... W^^' with the same coefGci- ente as occur in these combinations of V,, ..., V,', we have a set of t' integrals «?„, .... Wi^, such that tVir = KWir+ Wr, (r = 1, ..., t'). These integrals are linearly independent of one another, and also of Wj,..., Wr, before obtained. They constitute the aggregate of linearly independent integrals of this type ; for if there were another linearly independent of them, it would imply that A^ (li) had t' + 1 elementary divisors instead of only t'. As regards the two sets of integrals already obtained, it may be noted, (i), that the set TFi, ..., W^ can be linearly combined among themselves, without affecting the characteristic equation WJ^kW,.: yGoosle 21.] INTEGRALS 59 (ii), that to aach integral of the set Wn, .... Wi,< there may be added any linear combination of the integrals of the set Wj_, ..., W^, without affecting the characteristic equation If the index of each of the elementary divisors of Ai(il) is unity, then t'—ct^t, so that the number r + r of integrals obtained is then equal to <t, the multiplicity of the root of A (Xi) = in question. In every other case, ■/ + t< a: 22. When t' + t < tr, so that t' is less than the degree of ^i(il), we use the integrals w„, ...,w,,/ to modify the funda- mental system W^, ..., W^, w^^j, ...,Wm, substituting them for w,+,, .... «!,+/ in that system. When the variable z describes a simple closed contour round a, the effect upon the elements of the modified fundamental system is to change them into IJV, .... W,', Wii, ..., w'it', w\^i. ..., wj,, where t ^-t —\, and w; = kW„ w/-7ftF,-l-... + 7„V. + 7t,T+iW„ + ... + 7tAWi,' + 7(,A+,WA+,+ ... +7i,mW,„, for r = 1, ..., t; s = 1, ..., t'; f = \+ 1, ..., m. The fundamental equation derived through this system is A (fl) = {« - Iiy+''^, (11) = 0, where J.j(Il)=l 7x+,,*+, -ft, 7A+i,*+si ■■-. 7*+i,™ I- 17™,''+! . 7™,^+^ . ■■■, 7m,m-ft| Also * is of a root of A^in,) of multiplicity <t — t--t'. By a further application of the proposition (§ 20) connecting the elementary divisors of A (il) and -4, (H), the indices of the elementary divisors of ^^{n), which are powers of k — CI, are seen to he ff — (7] — 2, (7j — (Tj — 2, (T^ " o^s — 2, .... say t" in number. The procedure from the equation A^ (fl) = to the corre- sponding sub-group of integrals is similar to that adopted in the case of the equation .A, (H) — ; and the conclusion is that there y Google 60 COMPOSITION OF A GROUP [22. exists a sub-group of r" integrals w^i, iCs,, ..., Wj^", characterised by the equations for ( = 1,2, ...,t". And so on, for the sub-groups in succession. Combining these results, we have the theorem* ; When a root k of the fmidammtal equation A (fi) = is of ■multiplicity/ <r, and when the elementary divisors of A (12) associated vjith that root are a group of a lineai ly mdtpendpnt iiitegt als is associated with that root : this group constat-' if a mimher /r — a-i of sub-groups, which satify the equatioii', Wf = KWr, for r=l, T, Wa = KW^ + w,t, for t = l, ..., t", and so on. The integer r is the number of elementary divisors of A (ii) ; t' is the number of those divisors with an index greater than unity ; r" is the number- of those divisors with an index greater than two ; and so on. The group of <r integrals, and in~ a- other integrals, all linearly independent of one another, make up a fundamental system : tlie m. — a other integrals being associated with the m — (T roots of ^(li)=0 other than D^—k. When these roots are taken in turn, wc have a single integral associated with each simple root, and a group of integrals of the preceding type asso- ciated with each multiple root, the number in the group being equal to the order of multiplicity of the root. We thus have a system of integrals of the original differential equation distributed among the roots of the fun<lamental equation associated with the " That pait ot the theoiem which establialiei the e^isteiice of the group of inteitralB ata ttated with a multiple loot is due to Fuoh'i Cielle t lwi (18fl6), p. lifb but the initial eipiession ti'O 'o 'he members of the group was much moie complicated The part which arranges the group m sub gioups each with its own chaioeteristic eijuafion is due to Hamburgei CrdU t Litvt (1873) p. 121 he takes it in an aiiangement \vhieh will be found in the next seotiou The association of the sub-groupR with the elementaiy divii^oic of i \Q] i& due to Canorati Compte' Kemiits, t sen (1881) p 177 y Google 22,] OF INTEGRALS 61 singularity : that the system is fundamental is manifest from the facts, that the initial system was fundamental, and. that all modi- fications introduced have been such as to leave it fundamental. Ex. 1. Two independent int^rala of the equation Hence when tho variable describes a sirajilc closed contour round the origin it) the positive direction, we have and therefore the fundamental equation belonging to the origin (which is a singularity of the equation) ia I -\~e, 1=0, I -2jrt , -\^e ■: that is, it is (fl+i)'.o. Similarly, two independent integrals of the equation „dho , dw „ are given by Hence aftei' a simple closed contour round the origin, we have where n is e^" ; the fundamental equation belonging to the origin is : -l_S, 1 = 0, I , a-6 ' thatia, Ex. 2. Consfruct the linear differential equation of the third order, having for three b early ndepe del t ntegra!'' btain the fundamental equation apperta n ng to the r g as a a gular ty ; and from the form of the diflere tal eq at nn ver ty P" car^a theorem (§14) that the product of the three r t's of th a t ndame tal e'j t n ia unity. y Google 62 hamburger's [23, Hamburger's Resolution of a Group of Integrals into SuB-oROUpa 23. In the case when the roots of the fundamental equation are all distinct from one another, the general analytical character of each of the integrals of the fundamental system in the vicinity of the singularity has been obtained (| 18). We proceed to the corresponding investigation of the general analytical character of the group of integrals in the vicinity of the singularity, when the group is associated with a multiple root of the fundamental equation. We have seen that the group of iiuearly independent integrals can be arranged in sub-groups of the form W„ W.„ .... W, ;. the members of eaeh sub-group being aiTanged in a line and satisfying an equation characteristic of the line. Let these be rearranged in the form* 1^1, Wn, Wu, w^, ... Tfa, W]s, Ws3, Wjs, ■-■ each of the integrals in the new line satisfies an equation, and the set of characteristic equations for any line is, in sequence, the same as for any other line, so far as the members extend. When any such line is taken in the form where the integer fi changes from line to line, the set of the characteristic equations is * These are Hamburger's sub-gconps; see note, p. 60. Their number is equal >o the nwnbei of elemental; diviBois of A (Q) connected with the multiple root. y Google 23.] SUB-GROUPS Let we have and therefore [(^ 'i-iria = log K -»)•]■ = «(.- [% (z — a)~"]' = «! {z — «)~°. Thus «, {z — «)~" is unaltered by the description of a simple closed contour round a; it therefore is uniform in the vicinity of a, but it cannot be declared holomorphic in that vicinity, for a might be a pole or an essential singalarity of u^ {z — a)"". Denoting this uniform function of 2 — « by -^i, we have u, = {z-aY^^. To obtain expressions for the other integrals, Hamburger* proceeds as follows. Introduce the function L, defined by the relation i.ilog(.-,.), then, after the description of a simple contour, we have L' = L + l. We consider an expression F(i)=j'=^.,+(''-^)t,-.i+('';V"-''+- where \ T ) (^-1-r)! r!' and the functions -^i,...,-^^ are uniform functions of z — a. Then if, for all values of n, we take 7/^^, = (z-ay^-APF, where the symbolical operator A is defined by the relation ^F^F{L + 1) - F{L} = F'-F, we have = (3 - a)''«''+^ (A"F+ A^-^'F) " Crdle, t. LJLXT! (1873), p. 122. y Google 64 GENERIC FORM OF [23. holding for all values of n. These are the characteristic equations of the modified sub-group ; and therefore we can write with the above notations. This is Hambui'ger's functional form for the integrals, 24. The integrals ii,, ..,, u^ are a linearly independent set out of the fundamental system ; and the system will remain fundamental if Mj, . . . , w^ are replaced by /^ other functions, linearly independent of one another and linearly equivalent to Mj, ,.., m„. A modification of this kind, leading to simpler expressions for the sub-group of integrals, can be obtained. In association with F, take a series of quantities, defined by the relations »', = ^„ f^ = 11^5 + 2-^.^L + y{r,L'', i^=«. = >^.+ (''->«-,i+(^2>.-.i^4-... Then we have AF=auV^_, + a,^v^_^ + a„v^^,+ ... +a,,^_,v,. where the constants a are n on- vanishing numbers, the exa«t expressions for which are not needed for the present purpose. Then (e - ayv, is a constant multiple of (ir - ayAi^-^F, that is, of u, ; and it therefore is an integral of the differential equation. By the last two of the above equations, (s — aYv^ is a linear combination of (£— a)°A''~'F and (z — ayA''~'F, that is, it is a linear combination of Jia and Mi ; it therefore is an integral of the differential equation. By the last three of the above equations, (2 — ayv^ is a linear combination of (e - afA'^-'F, {z - afA^-'^F, {z - a)''A»-'^; that is. y Google 24.] A GROUP OF INTEGRALS 65 it is a linear combinatioa of u,, Us, Mj ; it therefore is an integral of the differential equation. Proceeding in this way, we obtain /j. integrals of the form {^~ay,„ {z-aYv, (^-a)-«,. Moreover, these are linearly independent, and so are linearly equivalent to it,, ,.., m^; for, having regard to the expressions of AF, ..., A«~'f, we see at once that any homogeneous linear rela- tion among the quantities v-^, ...,% would imply a homogeneous linear relation among the quantities F, AF, ..., A''~^F, that is, among u^, ..., u^; and no such linear relation exists. Hence Hamburger's sub-group of integrals is equivalent to {and can be replaced by) the sub-group {.~o)-.„ (^-.)-., (^-o)-»„ Accordingly, we now can enunciate the following result as giving the general analytical expression of the group of integrals, associated with a multiple root « of the fundamental equation*:-— Wke}i a root k of the fundainental equation A{d)—Q is of multiplicity a, the group of <t iniegrals associated •with that root can he arranged in avh-groups ; the number of these sub-groups is equal to the number of elementary divisors of A($) which are powers of k — 6; the number of integrals in any sub-group is determined by means of the exponents of the elementary divisors; and a sub-group, which contains fj. integrals, is linearly equivalent to the /J. quantities (z-ay,,. {z-afv, (^-»)-.„ where 27ria = log k, and the ji. quantities y are of the form V, = yjr, + 2-^^L + f,L\ ' This form of expresHion for the gronp of integrals appears to have been given first by Jiirgens, CreUe, t. Lxsx (1875). p. 154. See also a memoir by Fuchs, Berl. SitzangiUr., 1901, pp. 34—48. y Google 66 SUB-GBOUP OF INTEGRALS AND [24. where L = „ .log(z — a), (^ ] denotes -. q r;~, . f"*"^ Stti ^' ' \ r j . {p.-\—r)\ t\ the jj. quantities i/^j, ;.., ■^^ are uniform (but not necessarily holomorphic) functions of z — a in the vicinity of the singularity. Dll'FEBENTIAL EqL'ATION OV LoWEE ObDEK SATISFIED BY A Sub-Group of Integrals. 25. The preceding form of the integrals in each sub-group of a group, associated with a multiple root of the fundamental equation, has been inferred on the supposition that the coefficients of the linear equation are uniform functions. It will be noticed that the coefficient of the highest power of L in each of the members of the sub-group is the same, being an integral of the equation, — a result which is a special case of a more general theorem. Moreover, it is of course possible to verify that each member of the sub-group satisfies the differential equation ; and it happens that the kind of analysis subsidiary to this purpose leads to the more genera! theorem above indicated, as well as to a result of importance which will be useful in the subsequent discussion of the reducibility of a given equation. We proceed to establish the following theorem* which is of the nature of a converse to the theorem just established : If an expression for a quantity u be given in the form U = tj}n + <f>n-,L + ^„^L^+ ... + ^ai"-' + ii>iL"-\ where L = ^ — . log (z — a), and each of the quantities is of the form (^ = (^ — a)' . uniform function of z~a, a being a constant, then u satisfies a homogeneous linear differential equation of order n, the coefficients of which are functions of z uniform in the vicinity ofz = a; moreover, du ^ 3"-'M dL' dL" '"' 9i»-' are integrals of the same equation and, taken together with u, they constitute a fundamental system for the equation. " Fiiclis. in the memoiv quoted on the pceeedins page. y Google 25.] ITS DIFFERENTIAL EQUATION 67 (It is clear that „,^^^ is a numerical multiple of ij>,, and that the coefficient of the highest power of L in each of the announced integrals is, save as to a numerical constant, the same for all ; it is a multiple of 0i, which is an integral of the equation.) It is convenient to make a alight modification in the form of u ; we take ■■■+("i')+'^""+ *'-'''"■ where so that the character of the functions i^ and their form (except as to a mere numerical constant) are the same as those of the functions i^. Further, no change, either in the property that dii_ dhi are integrals of the equation or in the property that, taken together with u, they constitute a fundamental system, will be caused if they are multiplied by constants : so that, if the theorem can be established for -n,, ..., m„_j, where 1 3"-'« , 1 ! 8"-i, , , , r y^,L'^^ + ir,L«-' the theorem holds for the quantities as given in the enunciation of the theorem. y Google 68 SET OF LINEARLY [26. 26. Merely in order to abbreviate the analysis, we take m = 4 ; with the above forms, it will be found that the aaalysis for any particular caee such as n = 4 is easily amplified into the analysis for the general case. Accordingly, we deal with quantities w, u,, u-i, u?,, where M = i|^j + S-^-jL -I- Z-^M + ir^L", If u can be an integral of a linear equation of the fourth order with coefficients that are uniform functions oi' s — a in the vicinity of a, let the equation be dz' ds^ dz^ ds Let the variable z describe a simple contour round a ; this leaves the differential equation {if it exists) unaltered, and so the new form of u is an integral, say u, where u'=Kf,+ BKf,{L + 1) + dKf,{L + ly + K^p■,{L + If, where k is the factor comnjon to all the functions ifr after the description of the circuit. As u and u' are integrals of a homo- geneous linear equation, so also is Hence v also is an integral, and it is given by v' = B {«V^a + 2«i^, (i + 1) + K^i (£ + If} + 3 [Kf, + Kf, (L + 1)1 + K^, ■ and therefore w, = ^ [ - I'' — w I = Mj + M, , ntegral. Hence w' is also an integral, and it is given by and therefore is also an integral. y Google 26.] INDEPENDENT INTEGRALS oS* Thus integrals are given by t, =■«!, ■w-t, =«5, which proves one part of the theorem, viz, that u, u,, u^, Uj are simultaneous integrals of the linear equation if it exists. 27. In order to estabhsh the property that w, Ui, u^, % con- stitute a fundamental system of the equation if it exists, a pre- liminaiy lemma will be useful; viz. if A, B, G, D be functions free from logarithms and if they be such that a simple closed contour round a restores their initial values, except as to a con- stant factor the same for all, then no identical relation of the kind aA + 0BL + yCD + BDL' = can exist, in which a, /3, y, S are constants different from aero. For let the simple contour be described any number, N, of times in succession ; and let / be the constant factor acquired by the functions A, B, 0, D after a single description of the simple contour. Then we should have the relation /"[aA + 0B{L + N) + yC{L + Nf + SD(L + N'f] = 0, and consequently the relation aA + 0B(L + N} + yG(L + Ny+?iD{L + Nf^O. valid for all integer values of JV". Consequently, the coefficients of the various powers of JV" must vanish : hence 0= SD. = 38Z)i + yC, = 3SDi= + 27CZ -I- &B, 0= hI)L'-\- yGL^ + l3BL + <xA, the last of which is the original postulated relation. From the first of these relations, it follows that S = then, from the second, that 7 = then, fi-om the third, that y Google 70 EQUATION SATISFIED BY [27. and so, from the original relation, that a = 0. The lemma is thus established. It may also be proved that, if A, B, C, D be functions free from iogarithma, and if they be such that a simple closed contour round a restores their initial values, except as to constant factors which are not the same for all, then no identical relation of the kind aA + ^BL + riCD + BDL^ = can exist, in which a, 8, 7, 2 are constants different from zero. The proof is left as an exercise. It is an immediate inference from the course of the lemma that no relation of the form a'u + ^Ma + Jli^ + B'Ui = can exist, in which a, ^', i, S' are constants different from zero ; for proceeding as before, it would require 0= a'-.|r„ = 3a' ■f 3 + 2/3>5 + 7>i, = n'->^, + ^'--^s + i-if-i + ^'-^1, which clearly are satisfied oniy if a' = ^' = 7' = 8' = 0. Hence there is no homogeneous linear relation among the quantities m, Mi, Ws, Ms ; and they therefore constitute a fundamental system for the linear equation if it exists. 28. If the equation exists, we must have and in the operator A, the functions P, Q, ii, S are to be uniform functions of z in the vicinity of a. Let Z denote any function of z with the same characteristic properties as ^], ^.j, if~a. "^4; then with such an operator A, we have A(^i) =iA2 +Z', A(2'i=) = Z=A.?+2L2' ■\-Z", A (^i') = i^AZ + ZL'Z' + %LZ" + Z"\ yGoosle 28.] A HET OF IiNITGGRAJ.S 71 where Z' , Z", Z'" are functions of the same characteristic properties as Z, that is, aa i/^j, i^^, -Jcj, i^^, and they are iree from logarithms. Now as A?i, = 0, we have A-.^. - 0. As A«,; — 0, we have ii|^,j + iA'^i + -l^-i' = 0, that is, As A«a = 0, we have A-f 3 + 2 (LAi^i 4- f /) + i^A--^, + 2i^/ + -./r," = 0, that is, by using the two preceding relations, As A((=0, wehave A->/rj + 3 (LAa/t, + ■^;) + S (Z^A-»/r, + 2if / + 1/^;') + i^Ai/Tj + ZL^; + 3i/f," + -^1" = 0, that is, by using the throe preceding relations, Ai^^ 4 3i^/ + 3-f /' + i/r,"' = 0. Thus there are four equations ; each of them involves the coeffici- ents P, Q, -R, S linearly and not homogeneously. The required inferences will be obtained if the equations determine P, Q, R, S as functions of s, uniform, in the vicinity of a. Now each of the functions -yjr is such that {z — ay-'yjr is a uniform function of 2 - a in the vicinity of a ; accordingly, let (z-a)--^f^=0^, (/t=l, 2,3,4), where each of the ^'s denotes a uniform function. Substituting (z — ay$i^ for -^1^ in each of the four equations, the factor (s — a)' can be removed after the dififerential operations have been per- formed ; and then all the coefficients of P, Q, R, S, and the term independent of them, are uniform functions of ^ in the vicinity of a. Solving these four equations of the first degree for P, Q, M, S, we obtain expressions for them as uniform functions of s — a in the vicinity of a. (In general, this point is a singularity for each of the expressions.) It follows that, for these values of P, Q, Jf, S, the four quantities Wj, u^, v^, u are integrals of the linear differen- tial equation of the fourth order. y Google 72 PROPERTIES OF A SUB-GKOUP [28. As already remarked, similar analysis leads to the establish- ment of the result for the general case ; and thus the theorem is proved. Corollary I. It is ao obvious inference from the preceding theorem that, when a group of integrals is associated with a multiple root of the fundamental equation, any (Hamburger) aub-gi'OTip, containing (say) u of the integrals, is a fundamental system of a linear equation of order n with uniform coefficients. Further, it is at once inferred that the n' members of that sub- group, which contain the lowest powers of the logarithm, constitute a fundamental system for a linear equation of order n' with uniforiu coefficients. Corollary II. Similarly it may be established that one (Hamburger) sub-group containing n integrals, and another sub- group containing p integrals, constitute together a fundamental system for a linear equation of order w-l-^ with uniform coefficients. And so on, fur combinations of the sub-groups generally. Jix. Prove that if the linear equation in w has a sab-group of n iiitegi'als which, ill the vicinity of a singularity a, have the form W3 = ./'3 + 21;'^Z;-i-^/',Z^ where 2ii2X = li>g(a-m), and oaflh of the functions^ is sueli tbat (s-a) "^ is uniform, where e^'^ is a multiple root of the fundamental equation with which the sub-group of integrals is associated, then if the linear equation for V he eonatructed, where that iineiir equation has a corresponding auh-group of n - 1 integrals of the where the functions ip are of tlie same character aa the functions tJt. y Google CHAPTER III. Regular Integrals; Equation having all its Integrals rbqular near a singularity. 29. The general character of a fundamentai system of integrals in the vicinity of a singularity has now been ascertained. For this purpose, the main property of the linear equation which has been used, is that a is a, singularity of the uniform coefficients ; the precise nature of the singularity has not entered into the discussion. On the other hand, the functions <p which occur in the integrals are merely uniform in the vicinity of a : no know- ledge as to the nature of the point a in relation to these functions has been derived, so that it might be an ordinary point, or a pole, or an essentia! singularity. Moreover, the index r in the expressions for the integrals is not definite ; being equal to s— ■ log 6, it can have any one of an unlimited number of values differing from one another by integers. Hence, merely by changing r into one of the permissible alternatives, the character of a for the changed functions ^ may be altered, if originally a were either an ordinary point or a pole : that character would not be altered, if a originally were an essential singularity. It is obvious that the character of a for the integral is bound up with the nature of a as a singularity of the differential equation, each of them affecting, and possibly determining, the other. Accordingly, we proceed to the consideration of those linear equations of order m such that no singularity of the equation can be an essential singularity of any of the functions 1^, which occur in the expression of the integrals in its vicinity. In this case, the functions (f>, which are uniform in the vicinity of y Google 74 REGULAR INTEGRALS [29. a and therefore, by Laurent's theorem, can be expanded in a series of pasitive and negative integral powers of ? — a converging within an annulus round a, will at the utmost contain only a limited number of negative powers. To render r definite, we absorb all these negative indices into r by selecting that one among its values which makes the function ^ in an expression {z-af,!, iiuite (but not aero) when e= «. An integral of the form if = (3 - ay [<^, + <^, log (3 - a) + . . . + <^, |log (2 - a)Y\ where 0^, 0i, ..., 0, are uniform functions having the point a either an ordinai-y point or a zero, is called* regular near a. When a value of r is chosen, such that {z — a)~''u is not zero and (if infinite) is only logarithmicaiiy infinite like Co + c, log (2 -«) + ... +c„ [log (s -»)}", the integral is said to belong to the index (or exponent) r: the coefficients c being constants and not all of them zero. Similarly, when the singularity a is at infinity, and there is an integral |^^, + t.'''g^+-+t-(logJ)j. where ■^„, if-i, ..., i^, arc uniform functions having s = i» for an ordinary point or a zero, the integral is said to belong to the index or exponent p. It will be possible later to consider one class of integrals that do not answer to this definition of regularity : but it is clear that regular integrals, as a class, are the simplest class of integrals, and that the first attempt at obtaining integrals would be directed towards the regular integrals, if any. Accordingly, we proceed to consider the characteristics of linear differential equations which possess regular integrals : and in the first place, we shall consider equations all of whose integrals are regular in the vicinity of one of its singularities in order to determine the form of equation in that n tj A Th m I ) P T name for a y Google AND THEIR EXPONENTS As subsidiary to the investigation, one or two simple properties, associated with the indices to which the fuQctioiM helong, will first be proved. If a regidar fimction u (in the present sense of the term) hdong to an index r avd another v to an index s, tken M-i-w belongs to the index r — s: as is obvious from the definition. If a regular function u belong to am index r, t/ien -j- belongs to Ike index r—^, To prove this, let J.(.-«)—[_M>-*.+(.-«)*;)<loi5(,-,.)}-+.*.pog (.-»))— ], SO tiat -T- can onlj belong to the index r - 1, if some at least of the coefficients of powers of log{2 — w) are different fTOm zero when z = a. These coefficients when s = a is substituted: they cannot all vanish, for then </)„, ip,, ..., ^ would vanish when s = a, so that o,, c,, ..., i!^ would all be zero, and then u would not belong to the index r. Thus -r- belongs to the indes r — 1. There is one slight exception, viz. when « is uniform and the index r is zero ; then -y- is also uniform, and it may even vanish when 2 = a ; so that, if •« were said to beloi^ to the index 0, -5- could be said to helong to an index not less than 0. Form of the Differential Equation when all the Integrals are regular near a Singularity. 30. As a first step towards the determination of the form of a differential equation that has all its integrals regular, we shall obtain the index to which the determinant of a fundamental system belongs. Let the system be w,, w^, ..., w„: and let the y Google 76 Ft >RM OF DIFFERENTIAL EQUATION [30. indices of the members be r„ r^ the determinant in the form ...., r ■,„ respectively. We take Cw/^tf,'"-' '«,"-. ofS 3^2, where G i s a constant. The : quantity % is a solution of an equation, a fundamental system for which is given by It is clear that, if w,, w^, ... are all free from logarithms, then %, Vs, ... are also free from them. If hovrever there bo a group or a sub-group of integrals associated with a repeated root of the fundamental equation, we may take (§ 23) Wi' = Ow,, w/ = Wi + 6w^, so that , _ d /w.J\ _ ^ A ^\ ' dz \W|7 dz\8 wj " thus 1), is uniform and therefore free from logarithms. Similarly, Ui and all the quantities used in the special form of the determ- inant are free from logarithms. The indices to which Vi, %, ... respectively belong are ra — r, — 1, r, — ri ~ 1, r, — r^ — 1, . . . unless it should happen that, for instance, r2 = rj. In that case, we replace Wa by w^ + av^i, choosing a so as to make the new integral belong to an index higher tlian r^ or r, : this change will be supposed made in each case where it is required. Again, the quantity m, is a solution of an equation, a funda- mental system for which is given by ' dz \Vi' ' ' ds \vj ' The index to which % belongs is r,-r,-l-(r,-r,-l)-l, =.-.-r.-l, and so for u^, ... ; that is, their indices are '*a — ^a — !> r^ — r^—l,.... And so on, down the series. y Google 30.] HAVING REGULAR INTEGRALS Hence the index to which = w, + (wi-l)(r,-r,-l) + (m-2){r,-r,-l) + .,. ... + l(r.-r.-.-l) so that, denoting the determinant of the fundamental system by A {z) as in § 9, it follows that, in the vicinity of the singularity a, we have A (s) = (2 - a)'-'+^=+-+^'»-!™'"'-^i a (s - a), where Ji is a holomorphic function of its argument in that immediate vicinity, and does not vanish at a. 31. This result enables us to infer the form of the differential equation in the vicinity of the singularity a. Manifestly, the equation is d'^w d'"-'w d™~^w where A is the determinant of the m integrals in the fundamental system, and A, is the determinant that is obtained from A on d'^-'Wg . ^, , d™w» . -, , for s = 1, . . . , in, by the column -, — , dz"'-" •' dz™ fors = l, ..., m. Now consider a simple closed path round a. After it has been described, A and A^ resume their initial values multiplied by the same constant factor, which is the non-vanishing determinant of the coefficients a (§ 13) in the expressions for the transformed integrals ; thus p^ is uniform for the circuit. Hence, when the expressions for the regular integrals are substituted in A and A„, all the terms involving powers of log {z — a) disappear. Moreover, A belongs to the index n+...+r™-Jm(m-l); and so far as concerns the index to which A^ belongs, it contains a column of derivatives of order k, =m — {i)i — k), higher than the corresponding column in A, so that A„ belongs to the index n + ...+r,.-im(m-l)-«. y Google 78 METHOD OF FEOBENIUS [31. Hence p^ belongs to the index — k and therefore, in the immediate vicinity of a. the form of j), is given by where, at a and in tiie immediate vicinity of a, the function Pk{z — a) is a holomorphic function which, in the most general instance, does not vanish when z = a, though it may do so in special instances. As this result holds for k = 1, ..., m, we con- elude that, when a homogeneous linear differential equation of order m has all its integrals regular in the vicinity of a singularity a, ike equation is of the form, in that vicinity, where P,, Pj, ..., P^ are holomorphic functions of z — ain a region round a that encloses no other singularity of the equation. Construction of Regular Integrals, ey the Method OF Fbobenius. 32. The argument establishing this result, which is due to Fuchs*, is somewhat general, being directed mainly to the deduction of the uniform meromorphic character of the coefficients of the derivatives of w in the equation. No account is taken of the constants in the integrals i and it is conceivable that they might require the esistence of relations among the constants in the functions P,, ..., P,„, Hence for this reason alone, even if for no other, the converse of the above proposition cannot be assumed without an independent investigation. The conditions, which have been shewn to apply to the form of the equation, are necessary for the converse: their sufficiency has to be discussed. Accordingly, we now consider the integrals of the equation in the vicinity of the singularity!. Denoting the singularity by a, we write z-a = x. PAz-a) = pr{x)=p., (r = l, ...,m); * CrelU, t. Lsvt (1866), p, 146. t The following method ia due to Frobciiius ; leferenMS will be given later. y Google 32.] FOR REGULAR INTEGRALS 79 SO ihat the equation can be taken in the form valiii in the vicinity of a: = 0. If regular integrals exist in this vicinity, they are of the form indicated in §§ 18, 24, the simplest of them being of the form wf = a^ 2 g„x° = 2 g^xi''^" say ; should this be an integral, it must satisfy tlie equation identically. We have D«-= |,7(<7-1)..,(» -m+ !)-»(„ -!)...(» -m+2)j),-.. . say. Here,y(i«, tr) is a holomorphio function of x in the vicinity of the iC-origin and is a polynomial of degree m in (j, the coefficient of (7™ being unity i so that, if it be arranged as a power-series in x, we have f(x.,).f,{„) + ^fA,) + x-M,) + .... where /„ (tr) is a polynomial in ff of degree m, and /,{o), f^ (/r), . . . are polynomials in cr of degree not higher than m—1. Then = !/.«'*'/('»- P + ») = iai'+-fj,/.(p + ») + y,_,/,(p + «-l) + ... + ?./,«). If the postulated expression for w is to satisfy the equation, the coetBcienta of the various powers of x on the right-hand side must vanish : hence o-i/./.W, -»/.((>) +?./.(p + i), O-S./.W + »/,((> + 1)+S./.(P + 2), and so on. These equations shew that the values of p, which arc to be considered, are the roots of the algebraical equation fM-o y Google 80 CONSTRUCTION OF [32. of degree m in p: and that, for each such value of ^, s. = -, _!h_ 7.(P + l)/.(P + 2)-'-/.(P + ») ""' vhere — h^ (p) is the value of the determinant /.(p + i). , , Me) . /.(p + i), /.(p + 2), ,-., , /.(p) /.(p + i). />(>>+2). /.(p+8) , Mf) /„(, + !), /.-.(p + 2), /„(p + 3) f,{p + ,-\), /,_,(p) /„((.+ !), /.-.(p + 2). /.-.<P + 3) /,((. + "-!). /.W SO that h, (p) is a polynomial in p. If no two of the roots of the equation /, {p) — differ by whole nnmbers, then no denominator in the expressions for the si coefficients (j„ vanishes ; the expression £r(«, p) is formally a for an integral, but the convergence of the series must 1; lished to ensure the significance of the e If a group of roots of the equation /„ (p) = differ among one another by whole numbers, let them be P. p + e^. , p + e, where the real part of p i-< the smallest, and that oi p + e is the largest, among the real pait** of these roots; equality of roots would be indicated by coiiesponding equalities among the positive integers 0, e^, ..., e. We then t^ke g,-Mf + i)...f.(.P + i)s, and thus secure that no one of the coefficients g„ becomes infinite. The condition, that the equation shall formally be satisfied, has imposed no limitation upon ^j, which accordingly can be regarded a'i arbitrary ■ hence tj also can be regarded as arbitrary. 33 I o ie to Jc 1 v th both sets of cases simultaneously, tl e 1 il exp es n s on t ucted in a slightly different manner. A ] a n etr c quant ty k s ntroduced and it is made to vary wt!n eg s round the oots of /o(p) = 0, each such region round a ro t be ng chose o is to c itain no other root. The quantity g n tl e hr t Set of c s and the quantity g in the second set, y Google 33.] REGULAR INTEGBALS 81 are arbitrary ; they are made arbitrary functions of a. Quantities rji, t^i, ... are determined by the equations -»,/.(« + !) + <;,/, (a), = ft/.(» + 2) + !7,/,(»+l) + <7./.W, the same in form as the earlier equations other than the first : these quantities g are functions of a. Moreover, we have ''■<''>' /.(»-M)/.(»'l1).../.(.+ .;) ''-<°>^ in consequence of the assumption as to g^ (a) in the second set of cases, and of the regions round the roots of /„(/)) = in which a varies, it follows that the quantities gi, g^, ... are each of them finite for all variations of a within the regions indicated. We thus have an expression y=g{x,a)^ 2 g.x-+''-, also = i^£r.x"+" /<*.« + ") = ^„(«)/„(»)^-, the coefficient of every power of x except x" vanishing, in conse- quence of the law of formation of the quantities g. 34. We proceed next to consider the convergence of the power-series for y, before bringing the equation satisfied by y into relation with the original differential equation. We denote by R the radius of a circle round the a;-origin within and upon which the functions pi, .... pm are holomorphic : so that the circle lies within the domain of this origin. Then/(ic, a) and its derivatives with regard to x are also holomorphic for values of x within the circle and for all values of a considered. As the first of them, say /' (x, a), is of degree in a. one less than f{a:, a), it is convenient to consider that first derivative : let M (a) be the greatest value oi\f' {x, a)| for the values of a; and a, so that, as y Google 82 CONVEKGENCE OF THE [34. we have* and therefore, as k + 1 is a positive integer ^ 1, also i/„<»)|«-B-if(«). By the definition of the regions of variation of a and the signi- ficance of the integer e, it follows that the quantity /o(a + j^ +1) is distinct from zero, for all values of w ^ e and for all values of a ; hence, as from the equations that define the coefficients g, it follows that < i/^(.^,^ + i) -| li.!;.l-S-'*(«)+ L?.|-B-+«(" + 1) + ... say, where 7^+1 denotes the expression on the right-hand side. Evidently 7.H.,i/.(a + »+ 1)1-7. !/.(«+ >')|J!- = l!;.|«"<« + ») « 7, i¥(a + »), and therefore Let a series of quantities F^ be determined by the equation jf(.+») 1 1 /.(«+^) 11 ii/.(a+rti)rj2|/.(«+»+i)Ir for values of I'^e; and let r, = 7,. Then all the quantities F thus determined are positive, and we have ls'.«l<7.t.<r.„. Consider the series r,if + r,+,a;-+>+... + r,a!- + ...; r.+,-rJ y Google 34.] SERIES FOR THE INTEGRAL 83 its radius of convergence is detennined* as the reciprocfxl of Lim -^ , Now M{d) is the greatest value of the moduhis of - {0 - 1)...(0 - m + 2) p,' - ... - pr^' within the circle ]aT| = 7i. As the functions p/, ...,p,„' Eire holo- morphic within the circle, there are finite upper limits to the values of [pi'l, ..., \pm'\ within the region, say M^, ..., M^; then say, where |^| = a. Again f,(e)=0(d-i)...(6-m + r)-$(e~r)...(0-m + 2)pAo)-... ...-P^m, ao that, if ^(»)--„" + ,r(, + l)...(» + m-l) + ,(,7 + l)...(,r + m~2)!p,(0)|+... + lp,(0)|, we have |/.(fl)|>i9-| -!/.(«)- 9-1 St"- !/.(«)- 9-1, and the term in 0™ being absent from /, (0) - 0'^, and the term in ff™ being absent from i^ (a-). Moreover, as these quantities are required for a limit when v tends to infinity, the quantities a- and will be large where they occur; thus o-™ is greater than (ff), which is a polynomial in a- only of degree m — 1. Hence j/. (9)1 S »--*(«). Returning now to the expression for r^^^ ^ F^ , let /3 denote |al; tlien |c< + ,+ l|S>, + l-/3, so tiiat |a + i.+ l|-S(v+l-(3r. Again, \a + v+l\tii + X+l3, so tliat *(|« + i- + l|)«*(» + l + /3), and therefore j/.(» + « + l)|>(« + l-»--*(..+ l+/3), * Chrjstal'e Algebra, vol. ii, p. 150. y Google 84 CONVERGENCE OF THE SEEtES [34. finally, \a + v\^v + 0, and therefore so that M{a+ v) ^{" + 0) \f,(_ci + v+l)\^iv+l-m'-'-<l>l^ + l + &y Now 1^ (rr) is a polynomial in <r of degree m — 1, as also is <j) (o-) ; hence, owing to the term (v + 1 — 0)"^ in the denominator on the right-hand side, we have «(«+» ) _ iiri/.(«+»+i)r ■ for all values of /9, that is, for all values of a within its regions of variation. Again, as /„ (a) is a polynomial in a of degree m, it follows that .../.(<. + « + !) ■ for all the values of a, and therefore Using these results, we have T . r,„ 1 and therefore the series converges within the circle ja;| = B and for the values of a : conse- quently also the series 7.ic' + 7s+i«'+^-l- ... converges for the same ranges of variation for x and a. The addition of a limited number of terms that are finite does not affect the convergence : and therefore converges, for values of x within the circle \a:\ — R, and for values of a within its regions of variation. Let any region for a be defined by the condition \a-~p\^r. Then the series converges absolutely within the a;-circle of radius It and the a-circle of radius r. Let R'<R, and r^<r; and let K, K denote any finite positive quantities which may be taken y Google 34.] INDICIAL EQUATION 85 small ; then* the series converges uniformly for values of x and a siicli that \M<«-.. |a-p;<r'-.-. Thus the scries converges uniformly in the vicinity of the ai-origin, for all values of a in the regions assigned to that parametric variable. By a theorem due to Weierstrassf, the uniform convergence of the series, which is a power-series in x and a function-series in a, permits it to be differentiated with regard to a; and the derivatives of the series are the derivatives of the function represented by the aeries within the a-regions considered. Significance of the Indigial Equation. 35. We now associate the factor x° with the preceding series, and then we have g (*■, «) = a;- S g^a:' = 2 «/.^+' as a series, which converges uniformly within a finite region round the ^-origin and can be differentiated with regard to a term by term. (It may happen that the origin must be excluded from the region of continuity of g (ic, a), as would be the case if the real part of a were negative ; the origin must then be excluded from the region of continuity of the derivatives with regard to a, owing to the presence of terms such as ^„a;°iog3:.) The function g{a), a) thus determined has been shewn to satisfy the equation iij,(=.,<.) = i-/.Wi,.(.). As associated with the original differential equation, this result requires the consideration of the algebraical equation {hereafter called the indicial equation) /.((■)=<> of degree tn. The preceding analysis indicates that two cases have to be discussed, siccording as a root does not, or does, belong to a group the members of which differ from one another by * Tlie uniform convei^ence with r^ard to a; is known, T. F.,% li,finn. The uniform Gonvergenoe with regard to a is eatabliahed bj means of a theorem due to Osgood, Ball. Anter. Math. Soc, t. iir (1897), p. 73 ; see the Note, p. 122, at the end. of this chapter. t Ges. Werhc, t. ii, p. 208; see T. F,. i%S2, 33. y Google 86 INTEGRALS ASSOCIATED WITH [35, whole numbers (including a difference by zero, so as to take account of equal roots). Firstly, let p be a simple root ot fo(p) = 0, in the sense that it is not equal to any other root and that the difference between p and any other root is not a whole number. Then when we take a =p, all the coefficients 3i, ^5, ... in 5- (a;, p) are finite ; we have that is, is an integral of the differential equation : it is associated with the simple root p of the equation fo(p)=0, and it is a regular integral. 36. Secondly, let />„, p,, ..., pn constitute a group of roots of /o (p) = 0, differing from one another by whole numbers and from each of the other roots by quantities that are not whole numbers ; and let them be arranged so that the real parts of the successive roots decrease : thus the real part of p^ is the greatest and that of p„ is the least in the group. In order to secure the finiteness of the coefficients^,, ^2, ..., it now is necessaiy to take S.(a)-/.(a + l)/.(. + 2) .../.(a + e)j(a), =/(«)<,(.), say, where e^p„ — pn, and ^(a) is an arbitrary function of a: and now Ds, (»,«)- I'S («) n /, (« + ,.)- rj (a) F(„), where j'W-^n !/.(« + .)). Further, there may be equalities among the roots in the group r let pu, Pi, pj, pic, ... be the distinct roots taken from the succession in the group as they occur, so that po is a root of multiplicity i, Pi of multiplicity j - i, pj of multiplicity k —j, and so on. Then in ^(a), there is a factor {a — p^y through its occurrence in /„(«); there is a factor {a — pi)', through the occurrence of (a — piV"^ in /o(a), and the occurrence of (a — p,)' in /(,(a + po — pi); there is a factor (a — Pj)*, through the occurrence of (a — pj)*~J iny„(a), the occurrence of (a — pj)^ in/o{a + pj — pj), and the occurrence of (a - pjY in /„ (a + p,- pj). Now 0<i<j<k<..., y Google 36.] A GROUP OP BOOTS 87 so that, for F(a.), p„ is a root of multiplicity i, that is, 1 <it least : pt is a root of multiplicity j, that is, i + 1 at least ; pj is a root of multiplicity k, that is, j+ I at least ; and so on. Hence if p, be a root in the group as arranged, it is a root of F(a) of multiplicity K + 1 at least ; and therefore r«>)i .„_ L 3»' J. -p. for /i=0, 1 K. But Be, (.«,«)=»!-<, (.)*■<«), and g{3:, a) can be differentiated with regard to a; hence = 0, for /li = 0, 1 , . . . , K certainly, and for all other integer values of /x. less than the multiplicity of p„ as a root of ^(a)=0. Conse- quently, the expression ^ r s^g(^,« )"| 9p/ say, for the same values of fi, provides a set of integrals of the equation. Moreover, each of the distinct roots in the group thus provides a set of integrals; we must therefore enquire how many of the integrals out of this aggregate are linearly independent. 37. We first consider the members of any set ; they are furnished by for a value p assigned to a, and for a number of values of fi, say 0,1, ..., K. Kow and therefore - + ...+(loga;)'Ssr,(a)x'|, y Google 88 SUCCESSIVE SETS ASSOCIATED WITH [37. where it will be noticed that the coefficient of the highest power of logic on the right-hand side is g {x, a). Hence the set is y, ^wlogx + w,, 1/2= w (log xy + 2^1 log x + w^, + icw^-i\ogx + w/,, where the coefficients Wp are independent of logarithms. From the fact that y^ contains a power of log a: higher than any occurring in yo,yi, •■•, ^p-i- it follows (by the lemma in § 27) that no linear relation of the form c,]/, + c,y, + ... + c^y^ = cao subsist among the integrals. 38. Nest, we consider the sets in turn, associated with the values pn, Pi, pj, ... of a, as ai-ranged in decreasing order of real parts. The earhest of thorn is given by a = p^ : and it contains the i members for /i = 0, 1 i-1. Now /.(«).(«- p.)' (« - Pi)'-' (■< ~ ft)" ■■■("- Pi)"*-' A . *-(.).(«-?.)'(«-?<)* (.t-nf ...(»-p,)"« A,. and therefore /<" -|lS) " '° - ''•>' <° ^ !'''>' ■ ■ ■ <° ^ l"^'^- where A,, A,, A^ are quantities which neither vanish nor become infinite for any of the values p„, pi, ..., pi of a Also g,(„).g(«}f{a). where g{a) is an arbitrary function of a; so that (/o(a) does not vanish for a = pn ; a^id therefore the various quantities derived y Google 38.] ROOTS IN A GROUP 89 from ^o(al for 'x = p„, including ^„{a) itself, given by 'l' ^'^^ fi = 0, 1, ...,1—1 do not all vanish. Further L 3-' J-f. which is one of the integrals ; as the quantities do not all vanish, this integral belongs to the index p, ; and the coefficient of the highest power of log a: is g(x, po)- The iirst set thus gives i linearly independent integrals obtained by taking fi — 0, 1, ..., i— 1 in the preceding expression. That which arises from /I = is where all the coefficients are finite : thus it is a constant multiple of y,*. + aA+i A, (p,) + ^'+^^2 (p„) + . . . , an integral that is uniquely determinate. Now consider the second set : it is given by a = pi, and it contains the members for^ = 0. ...,i-l, i,i+l, ...,j-l. The value of5f(a^, a)is As regards the first part of this expression, we note that all the coefficients gy(a.) for v=0, 1, ..., pa — pi— 1 contain the factor (a - pi)* ; and therefore all the derivatives y Google 90 FORM OF THE INTEGRALS [38. for /t= 0, 1, ..,, i— 1, vanish when a is made equal to pi, while they do not necessarily vanish for higher values of fi. As regards the second part of the expression for g («, a), we wiitc it in the full form when a=pi, this becomes which accordingly is an integral, and it belongs to the index p„, being free from logarithms. But it has been seen that the integral, which belongs to the index p^ and is free from logarithms, is uniquely determinate, being g (x, p„) ; hence the foregoing integral, being the non-vanishing part of g {x, a) when a is p,-, is a constant multiple of (/{x, po), say Kg{a;, p„). It might happen that K=Q. A similar result holds for the derivatives of g(a;, a), for the values /J. = l, ..., t — 1. Consequently, it follows that the integrals for )i — 0, 1, ..., *— 1, can be compounded from the integrals of the first set ; they are i in number, but they provide no integrals additional to those in the first set ; and thei'efore, without limiting the range of their own set, they can be replaced by the i integrals of that set. As for the remainder arising from other values of fi, they are ^ r I ^yiM a^+^(|og^) i ^"1^'^-+. . .+(log*y i g^ (p,) wA Lv=o 9^-^ ..=0 o/'i* .=0 J for /i = i, V + 1, ---,7 — 1- Now ».(«) = » (•>)<«-?<)'<"- ft)' ■■■ {"-eifA,. 30 that the quantities L 3»- J...; for the values s = 0, 1, ..., /t in any one integral, and for the valuea fj, = i,i + l, ...,j — l in the different integrals, do not all vanish. y Google 38.] IN THE SUCCESSIVE SETS 91 All these integrals therefore belong to the index pi, and they are j — i in number. Moreover, the original set of j integrals, com- posed of these j — i and of the replaced i integrals, was a set of linearly independent members ; and therefore we now have j — i integrals, linearly independent of one another and of the former set of i integrals. Thus our second set provides j — i new integrals, distinct from those of the first set; and each of them belongs to the index p;. The first of them is given by fi — i: L.=o 9pi* * .=0 dpi'-' J which certainly contains terms not involving log w; ii j — l>i, the second of them is ' y g'^^'g^ (pi) „ Ki + l)log.i^?|fc) .- + ...], which certainly contains terms multiplying the first power of log a;; if _^' — 1 > i + 1, the third of them certainly contains terms involving the second power of log x ; and so on. The third set among our integrals is connected with the value a = pj, and it is given by for fi^O, 1, ...,k-l. Now The coefficients g, («) contain (a — p^' as factor for all integers v which are less than pi—pj', hence the quantities vanish for /i = 0, 1, ..., j — 1, and are different from zero only for fi=j, j + l, .... k — 1, As in the case of the preceding set, the quantities \_dai- \ .(")- yGoosle 92 FORM OF THE INTEGRALS [38, for /i = 0, 1, ..., i — 1, are linearly expressible in terms of the i integrals of the first set ; while for fi = i, i + 1, ..., j — 1, they are linearly expressible in terms of the j-^i integrals of the second set, subject to additive linear combinations of the first set. Thus the integrals in the present set which are given by r3'9<^.j.)i for ^ = 0, 1, ..., ^ — 1, provide no integrals linearly independent of the i integrals of the first set and the j —i integrals of the second set ; the j integrals in this new aggregate are linearly expressible in terms of those in the old. Now the present set of integrals, for ;u.= 0,l, ..., j—l, j, j+1, ..., fc— 1, are linearly independent of one another; and therefore the integrals for /^=j' j+ 1. ■■■- ^'-1 are linearly independent of one another, of the i integrals of the first set, and the j —i integrals of the second set. Thus the third set provides k~j new independent integrals, given by the k — j highest values of fi. The first of them, determined by f^ = j, is which certainly contains terms not involving log a:; if A — 1 >j, the second of them, determined by f-=j+ 1, is *ft- r S 8!^(p)^ + 0-+l)logaM. which certainly contains terms multiplying the first power of loga;; if ft— 1 >^' + 1, the third of them certainly contains terms multiplying the second power of log te ; and so on. Moreover, it is clear that all these k~j integrals belong to the index pj. The law of the successive sets is now clear. The last of thena, determined by Oi = pi, contains the integrals ^^l for /i = f, i + 1, ..., n, which are linearly independent of one another and of all the integrals of the preceding sets already retained. All these integrals, being h + 1 — J, in number, belong to the index pi. y Google 38.] GENERAL THEOREM 93 The results thus obtained maj be summarised as follows: When the equation fu{p) = hat, a group of roots p„, pi, ..., pn, which differ from one anothei hy integers (including zero) and differ from all the other toots by quariiities that are not integers ; when also the distinct roots aie arranged in decreasing succession of real parts, so that p^ is a root of TnuUiplicity i, pi is a root of m/idtiplicity j-~i, pjis a root of niultiphoity k —j, and so on, where pi' pi' pj' ■■■ ***"* distinct from one another and are arranged in decreasing succession of real parts; then, corresponding to that group of roots, there exists a group of n + 1 linearly independent integrals which are regular in the vicinity of the singularity. This group of integrals is composed of a set of i integrals, which are given by r3;:s_(«^-| F^l. for fi = 0, 1, ..., i — 1, and belong to the index p„; of a set of j — i, which are given by for iJ- = i, i + I, --., ? — 1, and belong to the hidea; pi; of a set of k—j integrals, which are given by for fi =j, i + 1, ..., k—1, and belong to the index pj ; and so on, the last set being composed of n + 1 — I integrals, which are given by I 3«' J„,/ for fj. = l, I + 1, ..., n, and belong to the index p;. is in substantial agreement with that which A different proof is given by iFuchaf: briefly stated, it amounts to the establishmont of an integral w,, belonging to the index p,,, to the transforma- tion of the equation of order m by the substitution y Google 94 INDICIAL EQUATION [38. into a linear equation in v of order m- 1, and to the discussion of this new equation in a manner similar to that in which the equation of order m is discussed. Estiositions of the method devised l>y Fuchs will also be found in memoirs by Tannery * and Fabry t. 39. Ai! the integrals of the differential equation, which has the specified form in the vicinity of the singularity, are regular in that vicinity ; their particular characteristics are governed by the roots of the equation /„ (p) = 0, that is, p(p-l)..,(p-m + l)-p(p-l)...(p~m + 2)p,(0)-...-p,(0)=0, the differential equation in the vicinity of the singularity being of the form ax'" ,.„i ^ ^ ' dx"^-^ This algebraic equation is of degree m, equal to the order of the differential equation; it is calted| the indicial equation of the smgularity, and the function f{a;, p), of which fa{p) is the term independent of x. is called the indidal function. From the form of the integrals which belong to the roots p of the indicial equa- tion of a singularity, and those which belong to the roots $ of the (§ 13) fundamental equation of the same singularity, it is clear that the roots of the two equations can be associated in pairs such that When the roots of the indicial equation are such that no two of them differ by an integer, the roots of the fundamental equation are different from one another ; there is a system of m regular integrals, and the m members belong to the m different values of p. When the indicial equation possesses a group of n roots which differ from one another by integers (including zero), the corre- sponding root of the fundamental equation is of multiplicity n : there is a corresponding group of n regular integrals, the ex- pressions of the members of which in the vicinity of the singularity may (but do not necessarily) involve integer powers of log x. When a root of the indicial equation occurs in multiplicity «, * Ann. de 1':Ec. Norm., 2' 86v. t. iv (1875). pp, 113-162. + Thflse, Faculty des Sciences, Paris (188S). t Cajley, Coll. Math. Paperi, vol. xil, p. 398. The names adopted by Fuchs are determinirende Ftindameittalgleichung, anAdeterminirende Fuiictton, respectively. y Google ,39.] LINEAR INDEPENDENCE OF THE INTEGRALS 95 SO that the corresponding root of the fundamental equation occurs in a,t least multiplicity k, there is a set of k associated integrals, the. expressions of all but one of which certainly involve integer powers of log a;. 40. Having now obtained the form of integral or integrals associated with a root of the indicial equation f^ (p) = 0, we must shew that the aggregate of the integrals obtained in association with all the roots constitutes a fundamental system. First, suppose that the roots of the indicial equation are such that no two of them differ by an integer; denoting them by ^1, pi, ..., pm, and the m integrals associated with these roots respectively by wfj, ..., w^, we have where Pj (s — a) is a holomorphic function that does not vanish when z = tt. No homogeneous linear relation can exist among these integrals : for, otherwise, we should have some equation of the kind Writing ^, = e2Tip,^ (6- = l, 2, ...,m), so that no two of the quantities S,, ..., B^are equal to one another, we can, as in § "18, deduce the equation c,i?/w, + cA'tVii + . . . + c^Or/w^ = 0, for any number of integer values of r, from the above equation, by making z describe r times a simple contour round a. Taking the latter equation for 7"= 1, ,.., to — 1, the set of m equations can exist with values of c,, .,., o™ differing from simultaneous zeros, only if ' , 1 , ..., 1 =0, which cannot hold as no two of the quantities are equal. Hence we must have e, = = Cs= ... =Cm, and no homogeneous linear relation exists : the system of integrals is a fundamental system. y Google 96 LINEAR INDEPENDENCE OF [40. Next, suppose that the roots of the indicial equation caci be arranged in seta, such that the members contained in each set differ from one another by integers. With each such set of rcots a group of integrals is associated, the number of integrals in the group being the same as the number of roots in the set. It is impossible that any homogeneous linear relation among the members of a group can exist: if it could, it would have the form If Wi, ..,, Wn involve logarithms, then (| 27) the aggregate coeffi- cient of the highest power of log (z — a) must vanish ; in the case of each integral in which the logarithm occurs, this coefficient (§ 25) is itself an integral of the equation, and therefore wo should have a relation of the form where the quantities w^, ..., w^ belong to different indices, say p,., ..., ps, nw two of which are the same; and w^, ..., w„ are free from logarithms. Dividing by {z — df>, we should have an equa- tion of the form hAz-ay'~'''Pr(^ - «) + ■■■ + h>P, (^ - a) = 0, where Pr, ■-., Ps ^■'e holomorphic functions oiz — a, not vanishing when z = (i. No one of the indices py — p^\& zero ; no two are the same : and so the preceding equation can be satisfied identically, only \ibr = ■■■ = hs. We therefore remove the corresponding terms from h{Wi + , . . -1- h,nW„ = 0, and proceed as before : we ultimately obtain zero as the only : value of each of the coefficients b. If Wi, ...,Wm do not involve logarithms, the argument, above applied to Wr, -.-, w,, can be repeated: there is no linear relation. The initial statement is thus established. If the tale of the groups, the members of each of which are linearly independent among themselves, is not made up of linearly independent integrals, then an equation of the form c,w,-|-...-l-c™w„ = yGoosle 40.] THE INTEGRALS 97 exists. Equating to zero (§ 27) the aggregate coefficient ol' the highest power of logs that occurs, we have, as above, a relation of the form C^Wr + . . . + C,W, + CjilUp + . . . + C5W, + . . . = 0, where w^, ..., w, belong to one group, Wj,, ..., w^ belong to another group, and so on. Writing CrW,4-..- +CsW,= F|, CpWfp+ ... +C5W,, = Ifj, ... we have Tf, + Tr,+ ...=0. Now let 6-i, =6^'"'', be the factor which, after description of a loop round a, should be associated with W,; let 6^ be the corresponding factor for W^; and so on: the quantities 0i, &^, ... being unequal to one another, because TT,, W^, ... belong to different groups. Then, as in ^ 18, we deduce the equation ^,*ri + i9,*F, + ...=0, after \ descriptions of the loop; and this would hold for all integer values of \. As before, taking a sufficient number of these equations for successive values of \, we infer that F, = 0, Tr.= o, ... ; if these are not evanescent, they would imply relations among the members of a group, and so they can be satisfied only if c,,= 0=...=c„ 0^ = 0=. ..-c,, .... Remove therefore the corresponding terms from the relation CiW,+ ... + CmWm=0, and proceed as before : we ultimately obtain zero as the only possible value of each of the coefficients c. Hence no homogeneous linear relation exists; the system is fundamental. Some examples illustrating the preceding method of obtaining the integrals of a linear equation will now be given. Eai. 1. Consider the integrals of the equatioa* in the vicinity of the origin. To obtain a regular integral, we take * Tho equation is not in the exact form indicated in tlie tast. We have m - 3, ji, {0) = 0, and so a factor x haa been removed; alao ive have multiplied hy the factor y Google 98 EXAMPLES OF [40. substituting, we hare provided = c,(a2-l)-.o(«+l), 0=3Caa{a + 2)-2Ci(a + 2)-Co(a-2)3, the last holding for n. = 2, 3, . . .. The indicial equation is .(a-!) = 0, giving (simple) roots n = 3, a = 0, so that a f;ictor o + l can be neglected : the relations among the coefficients are equivalent to (a + 2n-l)ca„^j-C3„=0, (a + 2«.)Cs,,5-^„^.= -"-^;n- (-+3i^(f^2« + 2j' Firstly, consider tlic root u = 2. We have so that the int^ral belonging to the index 2 is say the integral is u, where Secondly, consider the root o = 0. From the original form of the relations, by the first relation ; and the second relation is then identically satisfied, leaving c^ arbitrary. Using the reduced form of the relations for the higher coefficients, we have and therefore the integral belonging to the index is On subtracting c^ii from this integral, the reruainder is still an integral, and it belongs to the index in the form (say) Thus the system of two integrals, regular in the vicinity of x = 0, is y Google 40.] THE GENERAL THEORY 99 This method of dealing with the root a=0 is not quite in accord witii the course of the general theory ; it happens to be successful because Cj is left arbitrajy. In order to follow the general theory, we note that the coefficient of Cj in the original difference-equation cootaina a factor a which vanishes for the present root. Hence, taking where A ia finite, and so on ; thus »-«--(i+.-!i)+'.'-"{>+„-f,+(.+Tf;-s-^j+-}+.-«('..). where Ji(i>:, a) ia a holomorphic function of 3: which, by the general theory, is finite when a = 0. According to the general theory, this quantity should give rise to two integrals, viz. Taking account of the value of c^, the first of thorn is zero, thus giving an evanescent integral. The second is or adding to this itit^ral ^Ca, which is an integral, we have C{l-x), thus giving 1 - a; as the integral. Sx. 3. Discuss iT! a similar manner the regular integrals of the equation a the vicinity of the origin : likewise those of the equation x(l-x)i'/'-{l + i3!+%3^)«^ + {'3 + Ss:-3^)w^0 a the same vicinity. Ex. 3. Consider the integrals of the equation n the vicinity of the origin. Subatituting the espresaion y Google 100 bessel's [40. provided for B = 1 , 2, 3, . . . ; these values give where Tis a holomorphic fimotioii otx which is fiaite when a = l. The indicial equation has a repeated root a = l; hence two regular integrals are The former is c^i^:, say the integral is u, where the latter is c„a:logx+c„3^, say the integral is e, where ii=a: log x + x^. Both integrals belong to the index 1 ; and one of them must contain a logarithm, since the index is a repeated root of the indicial equation. Ei:. 4. Consider Bessel's equation for functions of order zero, viz. Substituting ■u)=o^x''+c,x'^*'-+,..+efie^*''+..., we have the latter holding for p=i, 2, 3, .... When those relatioi value of iw is — '•- V (.+2)'^(.+2)-(.+4)' ■■•;■ The indicial equation is a*=0, so that o=0 is a repeated root; thus the integrals of the eqiuation, both of them helonging to the indes zero, are M- [S]„.- The firet. of them is in effect, J^{x), on making Co=l. The second is +«.fi-2i^,(i+«+2rS7iii(i+H5)-...). y Google 40.] EQUATIONS 101 Denoting this by Kf^ wheu Cu = 1 , we have where ■^ {p) denotes the value of -j- {log n {:)} when s -p. The two integralu, regular in the vicinity of ^ = 0, are J^ and K^. Ex. 5. Consider nest Bessel'a equation for functions of order b, via. Dw=x^vf' + xv/ ■\-{3^-n?)vi-=i!. Substituting an expression W = C„:C^-i-Ci3!^*^+ ... +c„x'+"+ ... in the equation, we have provided .,{(a + l}^-«^} = 0, for^=l, 9, 3, ... ; we thus have r .-0^ 3^ n The roota of the indicial equation are When n is not an integer, the oorresponding integrals are seen to he effectively J„, J.,„. When ji IB zero, we have a rei>eated root ; this case has heen discussed in the preceding example (Ex. 4), When n is an integer different from zero, the two roots belong to a group ; and for a — — n, the coeffi.cient of afl^ is formally infinite, so that we have an illustration of the general theory in §§ 36—38. We take the roots in order. Firstly, let a= +«. : then the integral is i (-!)■ W n(p)n(«+ii")' on taking e^ equal to ---— ^. This is the function usually denoted by J'„ ; and it belongs to the index «, Secondly, when a= -n, one of the coefficients becomes formally infinite through the occurrence of a denominator factor {a + ^nf-it?. Accordingly, yGoosle bessel's equation [iO. and then |(„+a«)^_„3;^.ri = C{(«+a«)^-«2j^"ri^--^— -,+ ... +{-!)■ (a + 2)=-ft= n {(« + 2r)=-^=} say ; and now Two integrals atiae through this root, viz. Por the first of them, we have SO that it provides no new integral. For the second of them, we have raj,,-] 1 2C, •■£Y-''l' "'(''-'^y) ir L8»J„— »'n(«-i),;,W n(p) ■ " say ; and -S?--*''"<'°l="''['-»(in) + -2l(„ + l)(» + 2)l.S--] +*^''A 'n(')'n'(wi '■■!'"■'+*'"+'■'-» "■»©" say ; so that the integral is M\+ W^, In W^, the part represented hy is a constant multiple of J„ and therefore can be omitted, owing to the earlier retention of ./„. Eejecting this part, and taking C=-l2''"'n(j!,-l), the integral be(;omes _/2Y%-E_fc-pzi)('"'Y' U .;. n(j,) w + (f)"J. n4n'(i'+,) '"°^-*'"-H'-'--»@"' which differs, only by a constant multiple of ./„, from the espre-swion given by Haniel*. * Math. AnH,..t. i (186fl|, pp. 469—471, auotod iu my Treatise on Difurentinl Equations, p. 167. y Google 40.] EXAMPLES 103 Ex, 6. Discuss in a similar manner the integrals of the equation a(l-a-)V + {l~(a + 6 + l)a'}w'-(t6u^=0. in the viciniti^ of ;k = 0, and a:=\ : indioating the form for the latter vicinity when a+6="l. This equation is the differential equation of the hypergeoinetric series F{a, b, I, (b). When, in Logendre's equation (l-^)^-3,^+y(y+l)».0, the indei)endent variable is transformed to a:, where 2=1 — 2j;, it becomes a:(l-a-)«;" + (l-2ar)V+i'(F + l)w=0, which is the special case of the above given by 6=^+1, a= -p. The integrals of L^endre's equation in the vicinity of x=0 and of a;=l, that is, in the vicinity of 5=1 and of b=-1, can be deduced from those of the hypergeometric equation ; the actual deduction is left aa an esercise. Ex. 7. Apply the general theory to obtain the integrals of j,^w"' - ^xhe" + Ixw' - 8w = 0, which are regular in the vicinity of x=0. En. 8. Consider in the same way the equation i)()«) = (l+^):cW'-(2+4ic)icW + (4 + 10a!)«w'-(4 + 12;»)«' = 0. Substituting for w the expression as in the earlier examples, we have provided C,>(« + a-l)(« + a-S)^ + C„_l(« + a-3)V'i+«-4)=0, for!i = l, 2, 3, .... The roots of the indicia! equation are 2, repeated, and 1, so that they form ft group the members of which differ by integers. Moreover, when = 1, the coefficient Cj, which is _ (a- 3)^(a-3) '-" aW-W ' is formally infinite ; for that root, we shall take o..C(.-l)'. Firstly, for the repeated root n = 2, we have W^%X-{\^{a-tfR{x,a% where ii is a holomorphic function of x which remains finite when a = a . The two integrals are y Google 101 EXAMPLES 0¥ [40. it is easy to sec that they are constant multiples of both of which belong to the itides 2, Secondly, for the root a=l, we take a^=C(a—l)^, and theo i)(w) = C(a-l)=(<.-2)=:C", """. -C{a- -1)' '^- -G 1)'(.^2)'{.- .■(- + 1) — '. — + (a-l)-8(»,.), where §(iK, a) is a holomorphic function of .r which remains finite when a = I. In connection with this expression, three integrals are derivable, viz. '«^-' m.,, [SL.- The first of these is C2x% which is 2C% : it is not a new integral. The second is which ia SCm^ -ICu^: it is not a new iotegra,!. The third is adding to it liOu^—ZiOui, the new expression is still an int^ral and is a constant mulfcijile of %, where which manifestly belongs to the iodes 1. Ex. 9. Obtain the int^rala of the equations (i) (l + a^3W-(Z + i3:^)x^" + (i + lOx^)3^'~{i + 12^-^)v> = 0; (i!) (l + 'Lv}3^af'"-(4: + 20x)xhci'" + {U + 72x)x^!i/' which are regular in the vicinity of the origin. Sx. 10. Consider tiie integrals of Ihn=xw"'-i.(aj+bjX+...)w" + {a^ + b^+...)ii/ + {aj + h^ + ...)ir/=0 in the vicinity of ^ = 0, the constant a, not being an integer. To obtain the regular integrals, we substitute provided for values of n greater than zero, no one of the quantities /j,,/,, ... being of degree in n greater than 2. y Google 40.] THE GENERAL THEORY 105 Tlie i-oots of the indicial equation are a=0, 1, 2-a,. Por a = 2-a[, the difference- equation determines coeificients c„, which lead to a series converging for valuea cf |^[ within the common region of convei^eni^ of the coefBcients of id", w\ i/>. Por «=0 or 1, the difference-equation holds for values of n greater than 2 or 1 reapcctively ; the only other conditiona are o 2 4- -H -0 fo a= Hi ^ + =0 for a=l T) e tl e d fle'en e equat o again detem nes coefBcents wh ch in each die leid to a Mr es thit on et^es w th n the -lame regio a. the aer e. that helongs to the esp nent 2 o Ea h of the latte tegnl b a holo no ph funct on f and tl e efore the three te^iaU of the equat on wh h are regula n the v c n ty of j. =0 are — one, a 1 olomo ph t n t on of v belong ng to the dex i ^e ond i ke vise i holomorph f net on of % belonging to the ndex 1 , and a th rd, belong ng t tl e ndex 2 ^ Ex. 11, Discuss the r^ular integrals of the equation in the preceding example, when a, is an int^er. Ex. 12. Prove that the equation has m — i integrali \^hich are holoiiiorphie functinui of % in the vicinity of t = 0, when Oj IS not an integer, the various coelhcients a,-i-b^a! + ,., in the difterentiil e:iuation being holomorphic m that vicinity, and discuss the regulai integrals when •/.^ is -va integer (Poinoare.) Ex. 13. Shew that the aeries F( r^ = 1+-^r-l- "(" + 1 ) _,.i. ya,p,,r,r, ; -^^^^ -^2 ! p (p + 1) ^ {^ + l)r (r + 1) "^■■■ satisfies the equation .(«,-,„)J+a,; and obtain the other integrals, regular in the vicinity of j; = 0. Verify that, when a =r, the form of the function F, say (?()), tr,«), satisfies the equation of the third order and indicate the relation between the two differential equations. y Google 106 KEGULAR INTEGRALS [41. Regular Integrals, fkee FJiOM Logarithms. 41. Alike in the general investigation and in the particular examples, it has appeared that the regular integrals are sometimes affected with logarithms, sometimes free from them. Thus if no two of the roots of the indicial equation differ by a whole number, each one of the integrals in the vicinity of the singularity is certainly free from logarithms ; if a root of the indicial equation is a repeated root of multiplicity n, then the first « — 1 powers of log a; certainly appear in the group of n integrals which belong to that root. When a root of the indicial equation, though not a repeated root, belongs to a group the members of which differ from one another by whole numbers, the integral belonging to the root may or may not involve logarithms r we proceed to find the conditions which will secure that every integral belonging to that root is free from logarithms. Let the group of roots be denoted hy p^^, p^, ..., p^,..., arranged in descending order of real parts, so that p, — p„, for k = 0, 1, ... fi — 1, is in each case a positive integer: and consider the root p^. in order to obtain the conditions under which every integral belonging to p^ shall be i'ree from logarithms. In the first place, p^ must be a simple root of the indicial equation. Assuming this to be the case, we know that the integral belonging to p^ is in the notation of § 38. If we further admit the legitimate possibility that, to this expression, we may add constant linear multiples of the integrals which belong to the earlier roots Poi Pi. ---i pi^-i »nd still have an integral belonging to the root p^, then, in order to secure that every integral belonging to p^ shall be free from logarithms, the . integrals belonging to the earlier roots must also be free from logarithms; hence, as further conditions, each of the roots p„, p^, ..., p^^, of the indicial equation must be simple. These conditions also will be assumed The full expression for the integral belonging to p^ is the value, when a= p^,. of the expression ■[J; KloS") 2 aTr£f'«' + --+('»g'»)' ^ S.- y Google 41.] FREE FBOM LOGARITHMS 107 in order to be free from logarithms, the quantities for (7 = 0, 1, ..., /x- 1, and for all values i' = 0, 1, ... ad inf., must vanish : and if these conditions be satisfied, the above expression will acquire the desired form. The conditions will be satisfied for every value of u, if ^„(a) contains («— p^)" as a factor. But (p. 81) g-(«) ^-(°) = rr /„^ 3„(«) /o(« + l)/o(« + 2).../„(a + l') "'• -*■ say; and ^o(a), which (§ 36) is equal to ?(«)/{«), contains {a — py,)^ as a factor on account of its occurrence in f{'^)\ hence it is sufficient that H,{a) should remain finite (that is, not become infinite) when o. = p^, for all values of v. Moreover, -ffo(o[)=l. Having regard to the equation by which ^^(a) is determined, we obtain the relation i/,/.(. + ^)+ir,_./,(. + — 1) + ... All the quantities /^{a + v—l), ..., /„(a) are finite for values of a that are considered; hence ff„/c(c[ + i') is finite if fl'„(=l), H^, ..., /ff_i are finite, and therefore, on the same hypothesis, H^ will he finite for all values of v, if it remains finite for those values of the positive integer v, which make p^+v& root of the indicial equa- tion /5(^)=0. These values are known; in ascending order of magnitude, they are Consider them in ascending order. We have R,. When V =/5^_i — p„, a single factor /.(« + .) in the denominator vanishes when a^p^,,; and it vanishes to the first order, because p„_i is a simple root of the indicial equation. Hence, in order that jff„ may be finite ibr this value of v when a=: pn, it is necessary that hy{pi,) = 0, when v = p^^i — p^; and it is sufficient that A„(p„) should vanish to the first order. y Google 108 REGULAR INTEGRALS [41. When V = p^^ - p^, two factors in the denominator vanish when a = p^\ and each of them vanishes to the first order, because p,_i and p^_j are simple roots of the indicial equation. Hence, in order that H, may be finite for this value of V when a = p^, it is necessary and sufficient that K{a} should vanish to the second order when a = p„i the analytical conditions are that when V — pi.-2~ Pii ^'Wd a — p^. When p = p^_3 — p^, then the three factors in the denominator vanish when a — p„; and each of them vanishes to the first order, because p^_,, /Jb-s, pn-j are simple roots of the indicial equation. Hence, in order that H^ may be finite for this value of p when a = p^, it is necessary and sufficient that hy(a) should vanish to the third order when a. = p^; the analytical conditions are that ^ ' da da^ when v = pn^, — p^ anil a,~p^. Proceeding in this way. we obtain the conditions for the successive values of :' that need to be considered : the last set is that ~/^ = 0, (<r = 0,l, ...,p-l), when V = pa~ Pk ^tid a = p^. Such is the aggregate of conditions for a = p^. We have seen that, in order to secure the freedom from logarithms of every integral belonging to p^,, every preceding integral in the set as arranged must similarly be free : and so wo have, in addition, all the similar conditions ioT p^_^,p„^^,...,pi, there being no condition for the simple root p^. When all these conditions are satisfied, every integral belonging to p^ is free from logarithms. Manifestly these conditions also secure that every integi-al belonging to the roots /?„_,, p^-i, ■■■, pi of the indicial equation y Google 41.] FREE FROM LOGARITHMS 109 is free from logarithms : (one integral, belonging to p„, is always unconditionally free from logarithms) : it being assumed that each of the roots />o, p], ..., ^^ is a simple root of the indicial equation. The conditions thus secure that, when each of the /i + 1 greatest roots in the group of roots of the indicial equation is simple, the fj,+ l integrals belonging to those roots respectively are free from logarithms. The preceding investigation is based upon the results obtained by Frobcnius, Cretle, t. LXxvl (1873), pp. 224—226. A different investigation is given by Fucha, Crelle, t. lsyiii (1868), pp. 361—367, 373—378 ; see also Tannery, Ann. de I'tc. Norm., t. iv (187(5), pp. 167—170. Ex. 1. A simple illustration arises in Ex. 1, § 40, for tlie equation 3!{2-a?)w" - {x'+Ax.\.'i,){{\ - x)w' -i'w)=0. With the notation of the text, wa have /.(.)-. (.-2), p,-% c-I, Pi-O: oonsidcr A„ i ;a)fora=p, = 0, v = Pa- -Pi- = 2. ?.(«)^ ^('■)^o('') ^/,('' + l)A('< + 2)' g.i") "a"^^^^"'"'' *=(")= ^^^2/1 ("+ ^>-'« <"+ ^^ = (4-a)(a + l)(a+2)a. The (one) condition m the j re ent t ise w that A (qWO when 0=0 : which minife^tlj is latished Ex. 2. IfthcrtotH f tho indicia! e juati m are different from one another, then the integrals whirl beloi g to them certainly possess terms free from logarithms. (Fuchs.) Ux.Z. Lebpo, pi p le the roota of tie mdiciil equation which form a group, the members difieimg by itc^ois and nc two being equal; and assume them ranged in descending ordei of icil 1 xrts. Denote po-p, by s—l; and form the eqmtion satisfied l> V.*,(--'.); y Google 110 then according ( tion in W has nc integrals of the c by logarithms. EXISTBKCE OF AN INTEGRAL 3 the indicia! equation for the singularity a. negative roiita or has n^ative roots which riginal equation in w are free from li^arithn [41. = of the equa- re aifected (Fucha.) En. 4. Shew that the integrals of the equations (ii) (iii) where q and 6 arc constants, free from logarithms, the integrals of the equation □ the vicinity of the origin. [They a Ki^-h'^f.] 42. If, instead of requiring (as in § 41) that every integral belonging to an exponent p^ shall be free from logarithms, when p^ is one of a group of roots of the indicial equation of the type indicated in § 36, we consider the possibility that there shall be some one integral free from logarithms, belonging to the exponent and belonging to no earlier exponent in the group as arranged, no such large aggregate of conditions is needed as for the earlier requirement. Thus it is no longer necessary to specify that po, ..., /3,^i shall be simple roots of the indicial equation; nor is it necessary to specify that, even if these roots are simple, the integrals associated with them are of the required form. The conditions that arise will be particularly associated with a. = p^; but they will be affected by modifications arising out of the possible multiplicity of />„, ..., p^^i as roots of the indicial equation. The detailed results are complicated : a mode of obtaining them will be sufficiently indicated by an investigation of the con- ditions needed to secure that some integral free from logarithms exists belonging to pi and not to pn, with the notation of ^ 36—38, Suppose that p^ is a root of the indicial equation of multiplicity i ; and let yi, ...,yi denote the set of integrals associated with p^, where the expression of y,+], for « = 0, 1, ..., i— 1, is given by --P-^l. y Google 42] FBEE FROM LOGARITHMS 111 It' Pi is a root of the indicial equation of multiplicity j — i, only the first of the set of associated j — i integrals can be free from logarithms: even that this may be the case, conditions will be required. Denoting that first integral by W, we have Now W certainly belongs to the exponent p;. Its expression, in general, involves logarithms ; but there is a possibility of obtaining a modification of its expression, so as to free it from logarithms, if we associate with W a linear combination of -t/^, ..., yi with con- stant coefficients ; and the modified integral will still belong to pi but not to /Jo. Accordingly, consider the combination where the constant coefficients A are at our disposal ; this gives ~ xp" 's i 2 !^ t+, , '' ,, (log ^y ^'-Jf; xA . (=i„^oj>=(i( ^'pi{t-p)r ^ dp,*" j What we require are the conditions that may, if possible, secure that no logarithms occur in this expression for U. The least aggregate of conditions that will secure this result is ; first, for all values of v, which secures the disappearance of (log*')'i for all values of m and n such that pi + n = p„ + m, as well as ^_Sp. 0, for p = 0, l,...,p„~pi—l, these conditions securing the dis- appearance of (log ic)*~' ; next, .■(.-1) ?'.»._ y Google 112 CONDITIONS THAT AN INTEGKAL BE [42. for all values of m and n such that pi + n = pt, + m, as well as for p = 0, 1, ,.., p„ — pi appearance of (log ic)'"^ ii! dpf — 1, these conditions securing the dis- ; next, = Ai_,g„ip,) + {i -2) A, 3».. "3S -.^ 2 -2), S-S. such that pi + n-- -P. + " ., aa well as for all values of m and ) for p — 0, 1, . . - , Pa — Pi — 1 y and so on. This aggregate is both necessary and sufficient. Manifestly any attempt to reduce it to conditions independent of the constants A would be exceedingly laborious, even if possible. The difficulty arises in even greater measure when we deal with the conditions that some integral belonging to p^, where fj, > i, and to no earlier index, should exist free from logarithms. 43. If we assume zero values for all the constants A^, ..., Ai in the preceding investigation, the surviving conditions are cer- tainly sufficient to secure the result that the integral exists, free from logarithms and belonging to its proper exponent: but the conditions cannot be declared necessary. The aggregate of this set of sufficient conditions is, in the case of pi, that the equation shall hold for «■= 0, 1, ..., i— 1, and for all values of n. As in § 41, it can be proved that all these conditions will be satisfied if the equation has a simple root equal to pf. Assuming this to be the case, then an integral exists in the form ■■i.[ij.... y Google 43.] FREE FROM LOGARITHMS 113 which is free from logarithms and belongs to pi (but not to p^) as its proper exponent. If pi is a multiple root of the indicial equation, the remaining integrals belonging to pi as their proper exponent are certainly affected with logarithms. Conesponding conditions, that are sufficient (but are more than can be declared necessary) to secure the existence of an integral, free from logarithms and belonging to an exponent p^ (but to no earlier exponent in its group), can similarly be found; they are inferred from the investigation in § 41. If the equation when n = p^^i — pn, has a simple root equal to p^; if the same equation, when n = p^_5 — p^, has a double root equal to p^ ; if the same equation, when n = p,^^ — p^, has a triple root equal to p,. ; and so on, up to the case of n = pc — p^, when the equation must have a root equal to p^ of Kiultipiicity fi. : then an integral exists, belonging to pp. as its proper exponent (and not to any of the exponents po, p,, ..,, p^^, and free from logarithms. If p^ is a multiple root of the indicial equation, the remaining integrals belonging to p^, as their proper exponent are certainly affected with logarithms. On the preceding basis, the identification of the integrals, belonging to the group of exponents, with the sub-groups as arranged by Hamburger (§§ 23, 24) can be effected. The aggre- gate of integrals in the group, which are free from logarithms and belong to their proper exponents, not merely indicate the number of sub-groups in Hamburger's arrangement but constitute the respective first members in the respective sub-groups. The general functional forms of the remaining integrals belonging to any exponent are (save as to a power of a factor ^Tn) similar to those which occur in Jiirgens' form of the integrals in a sub- group*. 44. Ill the practical determination of the integi'ala of specified equations, it sometimes"!" '^ convenient to begin with that root * In tliie connection, tha followii^ memoirs may be consnlted: Jiii^ena, CrelU, t. [JLXX (1875), pp. 150— 16B; SeMesinger, Grelle, t. oxw (1895), pp. 159— 169, 309—311. + As to this process, see the remarlia by Cayley, Coll. Math. Papers, t. viii. pp. 458—162. y Google 114 cayley's [44. among the group of roots which lias the smallest real part, instead of beginning with the root that has the largest real part, as in § 36, When the process about to be discussed is effective, it has the advantage of indicating at once the number of integrals associated with the group which are free from logarithms ; but it is not always effective for this purpose, and it does not determine the integrals that are affected with logarithms. The equations determining the successive coefficients g,, </j, ... in the expression in the method ot Frobenius are (§ 33) = ^«/„(a + K) +?.-,/,(« + «- l)+...+S'».A("). for m= 1, 2, .... Let a group of roots of the indicial equation /.(«) = 0. differing from one another by integers, be denoted by p^, p,, ..., rr, where a is the root of the group with the smallest real part ; and replace a by «■ in the foregoing typical equation for the y's. Then, whenever o- + » is equal to another root of the group, the equation in its given form ceases to determine ^„, as a unique finite quantity. It may happen that the equation is satisfied identically; in that case g^ is arbitrary, as well as g^. It may happen that the equation appears to determine ^„ as an infinite quantity : in that case, we modify ^o as in § 36, and Qn is determinate after the modification. As often as the former ease arises, we have a new arbitrary coefficient; if k be the number of these coefficients loft arbi- trary, then K is the number of different integrals, associated with the group of roots and free from logarithms. These integrals themselves are the quantities multiplying the arbitrary coefficients in the expression Ex. 1. As an example in which the process, of dealing first with the root of a group that has the smallest real part, is efifeotivc as indicating the y Google 44.] METHOD 115 number of integriils free frum Ic^arithms, consider the equation The indicial equation is easily found to be (p-2)(p-3)S(p-5} = 0, th t tt re t ly ill 1 nteRral 1: 1 n n t the exponent 5, free fml tlm th m b ml ntegral bei "ing to the exponent 3, nd th e w 11 "« ta ly be an t gial, b 1 Dgin t that exponent and aff ted th 1 gar th and tl re may be n t gi 1 belonging to the ex[K) ent f ee f oia 1 gantl n i di 1 t k th alue p=2 ad ub tt t = + + + 'f'+ th j^ t t th mm diate p pose we d i>t consulor powers higher thau fi m v, because p=5 is the root of the mdicial equation with the highest real pait The equationa for determming the succes'<ive toefficients 0=Cj.O+eo.O, 0-Cj(-2) + c,(2) + to(-l), 0-Cs.O + C2(-2) + c,(3) + Co(-l); from nhiL,h we see that to, <■„ tj lemam aibitrary 111 the other cociB- tioita tu^e expressible in terms of them CunsequeotlT, the equation haa three mtegiais free fiom lonaiithms helonginn tn 2, \ 'i as their respective propel expunents (The equation was conotructed so as to haii, for a fundamental system ; the system is easily derived by writing when the equation for y is ^{l + j)y"'-aS(7 + 83)y" + j'(294-36«)y'-s(74 + 9fe)y' + (90+120«)^ = 0, which can easily be treated by the general method of Frobenius.) Ex. 2. As an example in which the process is ineffective, consider the Taking, as usual, y Google 116 EXAMPLE we have provided (p+n for values «=: I, 2, .... [44. If instead, of beginning with the root p=3 fts in tho general theory (§§ 35, 36), we try p = 2, the equation for the coefficients c gives «(«-l)«.-(»-2)>0.-„ determiuing c, apparently as infinite. To modify this, we take the equation for Cj then becomes (p-l)(p-2)c, = (p-3)a0'-2)(?, which is satisfied identically, when p = 2. Thus c, rornaina arbitrary ; but Co=0, The integral which would be obtained is, in fact, that which belongs to p = 3; and the process is ineffective. There happens to be no integral belonging to p=2 (and not to p=3) free from logarithms. The .Tctual solution ia easily obtained by the general method of Frobeniua. "We have ir= (7.P ((p - 2) + ^''~_f : + (p - 2)2 (p - 3)= R [z, f,)} , where R{z, p) is a holomorphic function of 3 when p is either 2 or 3; and then For p = 3, we have the integral For p = 2, we have the two integrals M)3=ryn =Gz^JrCifi\(igs^ZCz\ The integral belonging to the index 3 is free from logarithms ; that which belongs to the index 2 is effectively 2^+2^ log i, which is affected with a logarithm, so that the index 2 possesses no proper int^raJ free from logarithms. y Google DISCRIMINATION BETWEEN SINGULARITIES Discrimination between Real Singularity and Apparent Singularity. 45. The singularity, in the vicinity of which the integrals have been considered, is a singularity of coefficients of the equa- tion da'" z — a di™"^ (z — a)™ and the indicoa to which the mtegrals belong are the roots of the indicial equation for s = a, which is 0-0; p(p-l)-(p-<n + l} + p(p-l)...{p- m+2)P,((j)+... .■•+-P.W-0. In general, tlie integrals of the equation in the vicinity of a cease to bo holomorphic functions of s — a; thus they may involve fractional powers or negative powers of z —a, and they may involve powers of log (s — a). When this is the case, a is called * a real singularity. If, on the contrary, every integral of the equation in the vicinity of a is a holomorphic function of s — a, then a is called an apparent singularity of the differential equa- tion. The conditions that must be satisfied when a singularity of the equation is only apparent, so that it is an ordinary point for each of the integrals, may be obtained as follows. Let Wi, Wa, ..., w^ denote a fundamental system of integrals in the vicinity of the singularity a ; and suppose that each member of the system is a holomorphic function of 2 — a in that vicinity, so that the singularity a is only apparent. Let A denote the determinant {§ 10) of this fundamental system, so that _ 1 d^^^tVi d''^^Wi dwi ^__ j _ dz'"~^ ' dz™~^ > ■•■> ^^ > j rf™~'w2 d^~^Ws dwi I dz"^'- ' ^2™-^ ' "■' "dz ' dw„ y Google 118 REAL AND APPARENT [45. and let Ar denote the determinant which results from A when the column " - j - ^_,' is replaced by —r-^ , (for s=l, ..., m). Then as every constituent in A, and A is a holomorphic function of ^ -a in the vicinity of a, both A^ and A are holomorphic func- tions oi z — a in that vicinity; neither of them is infinite there. But as in § 31, we have m-i' <-> "'>■ and some one at least of the quantities P^ (a) is not zero ; hence, for that value of r, A,(o) a(») is infinite, and therefore A(<.) = 0, or the determinant of a fundamental system vanishes at an apparent singularity. Moreover, as in § 10, we have \dA^_ Pi(^) ^ _ A(tt) dQ(z-- a) A dz z — a z— a dz ' where ff (s — a) is a holomorphic function oi z — a; whence where A\s & constant. Now A is not identically zero near a, for the system of integrals is fundamental ; hence A is not zero. We have seen that A (a) = 0, and A {z) is a holomorphic function of s — a; hence P, (a) m,ust he a negative integer, numerically greater than zero. This condition is required, in order to ensure that a is a singularity of the equation. As each of the integrals is a function, that is holomorphic in the vicinity of a, it follows that the respective indices to which they belong must be positive integers ; and therefore the roots of the indicial equation Mp-l)...(p-m+l) + p(p-l)...(p-m+2)P,(tt)+... ... -|-pP„_i(a) + P„(a) = must be positive integers. (When one of these is zero, then P^ (tt) vanishes.) Moreover, no two of these roots may be equal ; y Google 45.] SINGULARITIES 119 for otherwise, the expressions for the integrals that belong to the repeated root would certainly include logarithms, contrary to the current hypothesis. Accordingly, let the roots be p„ p^, ■■■,pm, a set of unequal positive integers which we shall assume to be ranged in decreasing order of magnitude : they thus form a single group the members of which differ from one another by integers. The integral belonging to pi involves no logarithm. In order that every integral belonging to p^ may involve no logarithm, one condition must be satisfied : it is as set out in § 41. In order that every integral belonging to ps may involve no logarithm, two further conditions must be satisfied ; they are as set out in § 41. And so on, for each of the roots in succession until the last; in order that every integral belonging to p^^ may involve no logarithms, m — 1 further conditions must be satisfied, being the conditions set out in § 41. The aggregate of these conditions, and the property that the roots of the indicial equation are unequal positive integers, give the requisite character to the integrals. The condition that P](a) is a negative integer makes a a singularity of the differential equation. When all the conditions are satisfied, the singularity is apparent. In all other cases, the singularity is real Ex. 1. Consider whether it ia possible that ,r = sliouid be apparent singularity of the equation where k and X are constants. The first condition, that Pi(w) should he equal to a negative ii satisfied : in the present instance, it is -4. To discuss the integrals only ; and substitute : then I)v! = Ca{a-i)(a-l)x- .,(a + ™-4)(a + «-l)={X(« + «-l) + ^}c„_i, The indicia! equation, being (a-4) (a- 1) = 0, has all its roots positive integers ; so that another of the conditions is satisfied, roots form a group. equal to The two y Google 120 EXAMPLES [45. The integral, which belongs to the (greater) root 4 as its index, is a holo- morpbic function oi s: ; it is easily proved to be a constant multiple of (say) u, where "-^t*^ 1.4 ^+ 1.4 ■ 2.5 ^+ 1.4 ■ 2.5 ■ 3.6 ^^-j = a^(l +71^4-723^ + 73^ + . ..), for brevity. As regards the other root given by a=l, we have to assign the conditions that the integral whioh belongs to it contains no logarithms. In accordance witli the results of § 41, wo seo that there will be a single condition ; expr^sing it in the notation there used, we write Po=^< Pi = l' '•=Po-Pi=3. f=li and we have to find h^{a) for i-=^3, n=p,'=I. Now (§ 38) /o(a) = (o-4)(a-l), f^ Mffoi") /,(a + l)/„(a + 2)/,[« + 3)' so that A,(fl) = {X(a + 2) + ,c}{\(a + l) + «}{Xa + «}, Tho sole condition is that and therefore we must have K=-\, or -2X, or -3\. If K has any one of these values, the origin is only an apparent singularity of the equation. If K= -X, the independent integi'al belonging to the root 1 is If K= -2X, the integral is If « = - 3X, the integral is In all other cases, the origin is a real singularity of the difierential equation. The result, as to the relations between X and /i, can be verified inde- pendently. As w and u are solutions of the differential equation, we have 9M = T v/'-mi"=(^ + \\{wv/-mi'), and therefore where ^ is a constant. Hence dw \uj H* (l + yj_^+yiX^ + ys^ + ...f' y Google If every integral is to be holomorphic in the vicinity of the origin, it is easy to see that, as M belongs to the indes4, the only condition necessary is that the coefficient of - on the right-hand side should be aero. Thus wliich, on substitution for yj, y,, y^, and mnltiplication by — 36, givM thus verifying the condition obtained by the general method. In this example it appears that the integral, which belongs to the smaller root of the indioial eq^uation, is, in each of the three possible instances, a polynomial in ix. It must not be assumed that such a result always holds when a singularity is only apparent ; this is not the case*. j&. 2. Prove that the origin is aji apparent singularity for the equation where X and /i are constants ; and shew that no integral, holomorphic in the vicinity of the origin, can be a polynomial in ;c unless ^ is a positive integer multiple of X. &. 3. Prove that 3=0 and z= 1 are real singularities for tho equation ?(l-j)w"+(l-2s)w'-|)i'=0; and that 2 = 1, := -1, are real singularities for when n is an integer. Ex. 4. Shew that s=r» is a real singularity for every equation of the where /i {^} denotes a rational function of s. £:e. 5. Shew that, if £=<o be an apparent singularity for each integral of the equation where P and Q are holomorphic functions of £~' for lai^e values of |«|, then, if !Pf-j = X-l-negative powers of 2, -«©-- * See some remarks by Gayley, in the memoir quoted on p. 113, note. y Google 122 EXAMpr,Es [45. X must he a positive integer equal to or greater than 2, and. ft must be a positive integer which may be zero. Shew also that, if X = 2, then it must be Are these conditions sufficient to secure that each integral of the equation is a holomorphic function of 3"' for large values of \z]'i Ex. 6. Verify that every integral of the equation is holomorphic for lat^e values of \z\. Note on § 34, p. So. To establish the uniform convergence of a aeries ^.g^^" for values of n, Osgood shews {I.e., p. 85) that it is sufficient to h^ve quantities jl/„, indepen- dent of a, such that provided the series SJf„ converges. Take a circle in the a-plane large enough to enclose all the regions round the roots of /(p) = given by |n — p|"<j-' — k' ; and let this circle be of radius j-j, BO that r^ is a constant independent of a. With the notation of § 34, take constants C^, for values of i- ^ t, such that f (fi + O while G^^T^ = y^. Then, as ^{v, + v)>M{a-^v\ (-,■, + .)•"-,(,(.■. + .)< |/„(« + . + l)|, for all values of v. Now, as in § 34 for the ratios of the r's, we find and therefore the series converges, li! being leas than It. Accordingly, by taking the uniform convei^enee of the series ^ff^^ is established. y Google CHAPTER IV. Equations having their Integrals regular in the Vicinity OF every Singularity (including Infinity). id. We have seen that, if a linear ditfereiitiai equation is to have all its integrals regular in the vicinity ot any singularity a, it is necessary and sufficient that the equation should be of the form cfe™ J— (I rfa™"' {z — af dz™^ " {z — a)™ in the vicinity of that singularity, the quantities P,, P^, ..., P,„ being holomorphic functions of s - ra in a region round a that encloses no other singularity of the equation. We can immediately infer the general form of a homogeneous linear differential equa- tion -which has all its integrals regular in the vicinity of every singularity of the equation, including .2=00. As Fucha was the first to give a full discussion* of this class of equations, it is sometimes described by his name; the equations are saidf to be of Fachsian type or 0/ Fuoksian class. Let a,, «£, ..., ttp denote all the singularities of the differential equation in the finite part of the ^-plane, and write then the conditions are satisfied for each of these singularities by the equation d"'w_ S Qi, d'"'~'w dz™ g=x 1^' dz™"" * See his memoir, Crelle, I. lxvi (1806), pp. 139—154. + Care mast be eiercised in order to discrirainate betweea eq'iiitiuas of Fuclisian type and Fuehsian equations. The latter arise in eonneetion WLtb automorpMc funotions and differential equations having algebraic coefficients ; aee Chap. x. y Google 124 EQUATIONS OF [46. provided the functions Q, are holomorphic functions of z every- where in the finite part of the plane. To secure that the integrals possess the assigned characteristics for infinitely large values of z, we note that + = ..i!Q, where iJ is a polynomial in — and is unity when a = qd , and therefore = £P"ii" ©--.©• where /ii is of the same polynomial character as B., and is unity when 2 = CO . Now suppose that, for very large values of z, the determinant A (s) of a fundamental system helongs to the index o-, so that A(.)=^.r(l), where 2" is a regular function of - which does not vanish when 2 = CO . Then, with the notation of S 31, we have A.(»)=^— r.g), where T^ is of the same character as T, save that it may possibly vanish when z= k: taking account of the latter, we have A.W-^~ir.'(i), where e is an integer ^0. Thus ..4='—©. where U is a. regular function of - which does not vanish when 3 = 00 ; and therefore Q.-P.r y Google 46.] FUCHSIAN TYPE 125 for very large values of z. But Q^ is a holomorphic function of z near z = oo ; this property, imposed on the preceding expression, shews* that Q^ is a polynomial in z, of degree not higher than (p-i).. Moreover, it was proved in the last chapter that all the integrals of the equation are regular in the vicinity of a = a, when the quantities Pi, ..., Pm are holomorphic functions of s in that vicinity. Applying this proposition to each of the singularities (including co ) of the equation with the restriction upon Q,, ..., Q„ as polynomials in s of the appropriate degrees, we infer that all its integrals are regular in the vicinity of each of the singularities (including x ). Combining the results, we have the theorem, due to Fuchs-f: — When the m integrals in the fundamental system of a linear homogeneous equation of order m. liave a^, a^, ..., ap as the whole of their possible singularities in the finite part of the z-plane ; and when, all the integrals are regular in the mdnity of each of these singularities, as well as for infinitely large values of z; the equation is of tfie form d^_G^ d-^-'w G^,p^ d"^w G'™,p_i) de™' -^ ds'"'^ -yjr^ rfs"'~^ ' 'i|r'" where -^ denotes ti (z — a^,and, Oi^if,-!,, for /i = l,2, ...,m,is a polynomial in z of degree not higher than fi(p — 1). Conversely, all the integrals of this differential equation are everywhere regular, whatever be the polynomials G and -i/r of proper degree. Accordingly, this is the most general form of linear equation of order m, which is of Fuchsian type. ' This result may also be obtained by using the trausformation zx — l and applying to the equation, traasformed by the relations in g 5, the proper conditions for the immediate vicinity of a; = 0. t Crdie, t. i.xvi[186B),p. 146. y Google 126 EXAMPLES OF EQUATIONS [46. Ex, 1. Legendre's equation is (l-32)W-2sju'+m(n+l)ii'=0, say Its form siatiafiea alt the ueoesaarj conditions ; hence its integrals are regular in the vioinity of i=l, z= —1, and are regular also for inflnitely large values of 2. Similarly, the hypergoomctric equation, which is .(l-.).J- + {,-(. + S + l).)«'-a/5».-0, has all its integrals regular in the vicinity of ^ = 0, ! = 1, and regular also for infinitely large values of z. Eessel's equation of order zero is __1 ,_£ - J «* ^"'■' its integrals are r^ular in the vicinity of s=0 ; but, on account of the order of the numerator of the coefficient of w in its fractional form, they are not regular for infinitely large values of j. The same result as the last holds for which is JJesael's equation of order «. A form of Lanie's equation, which proves useful {see Chap, ix, §§ 148 — 151), is where A and B are constants ; its integrals are regular in the vicinity of any point in the finite part of the 3-plane congruent with 3 = 0, and these are all the singularities in the finite part of the plane ; but they are not regular for infinitely large values of s. Ex. 2. The sum of all the exponents associated with all the singularities (including oo ) of the equation of Tuchsian type obtained at the end of the preceding investigation is the integer |(p- l)ra{m — 1), a result first given by Fuchs*. The proof is simple. The polynomial ^p-i is of order not higher than j) — 1 : say G„-i = A^-^ + .... The indicial equation for the singularity a„ is ji'^)/]C/9_n...(S-m + 2) + ..., ' Crdle, t. Livi, p. 145. y Google 46.] OF FUCHSIAN TYPE 127 the unexpresBed terms on the right-hand aide constituting a polynomial in 6 of order not higher than m — 2. Hence the sum of the indices for the singu- and therefore the sum of the indices for all the singularities Oj, a^, ..., (tp in the finite part of the plane -*'-("-"+ iT?sr .Spm(»-1) + J, because Oj, a.2, ••■, ^p are tha roots of i^ = 0. The indices for co are obtainable by suiistituticg the indicial equation for t» is {-■irpij>+l)...{p+m-l)={-l)'-'Ap{p+l)...{p+m-2) + ..., so that the sum of the indices for co is -im(m--l}-A. The total suqi of all the indices is therefore Hf-l)M(M~-i}. Ex. 3. The general eqtiation of Fuchsian type, which haa all ite integrals regular in the vicinity of every singularity (including to ), has been obtiiined. The limitations upon tlie form of the type are mainly as to degree, so that generally the construction of the equations, when definite singularities and definite exponents at the singularities are assigned, will leave arbitrary elements in the form. The instances when the equations are made com- pletely determinate by such an asaignment are easily found. Taking the equation as of order m, we have polynomials o,_,W, o,_,(.) ff„-.(.) which, in their most general form, contain p-f-(2p-l) + (3p-2) + ...(mp-m-Hl) = ip™(™-i-l)-^m(™-l) The assignment of the positions of the singularities merely determines V' : it gives no assistance to the determination of the constants in the poly- nomials ff. Each of the /> singularities in the finite part of the plane requires ™ exponents, as does also the point £ = ai ; so that there are ™{p-|-l) constants thus provided. But, by the preceding example, their sum is definite : and thus the total number of independent constants thus provided is yGoosle 128 EQUATIONS OF FUCHSIAN TVPE [46. If therefore the equation is to bo made fully determinate bj the assign- ment of these constants, we must have and therefore ipm(™-l)=J(m-l)(M+3). When m=l, p can have any value; that is, any homogeneous hnear equation of the iirat order, which has ita integral r^ular in the vicinity of each of its singularities and of s = oo , is completely determined by the aaaign- ment of singularities and of the exponents for the integral in the vicinity of the singularities. For such equations of the first order, let a,, ,.,, Op lie th ng 1 t a the finite part of the plaoe ; let mj, ..., m^ be the ind to wh h th integral beloi^s in their respective vicinities, and let be the d x f 2=00, so that m+ 2 ^^ = 0. The equation is which gives the indes for : = so as equal to - 2 m,. , beiiig it* proper value. ■yiTien tn>l, then ao that, as p is an int^er, m must be 2 and then p=2. Thus the only homo- geneous linear equation of order higher than the first, which is of the Fuchaian type, and is completely determined by the assignment of the singu- larities and of the exponents to which the integrals at the singularities belong, is an equation of the second order : it has two singularities in the finite part of the plane, and it has a = co for a singularity ; and the sum of the sis indices to which the integrals belong, two at each of the singularities, is ^(2-1)2 (3 - 1), that is, the sum is unity. The discussion of the determinate equation of the second order of the foregoing type will be resumed later (§§ 47 — 60). IiTole. If p =0, so that the equation has no singularities in the finite part of the plane, the coefficients are constants if the equation is to be of Fuchsian tyX>e. The only singularity of the integrab is at co . If p=l, m>l, the number of arbitrary constants is less than the number of constants, due from the assignment of the indices at the finite point and at 3= CO : the latter cannot then all be assigned at will. For values of p greater than 1 and for values of m greater than 1, the number of arbitrary constants in a linear differential equation, which are left undetermined by the assignment of the singularities and their indices, is = ipm{m + l)-im{«,-l)^{m{p + l)-l} =i(™-l){™(p-l)-2}, which for all the specified values of p and m, other than m = 2 and p = 2 taken simultaneously, is greater than zero. y Google Ex. 4. Consider the equation, indicated in the Note to Es, 3, all whose integrals arc regular at the only finite singularity, which can be taken at the origin, and regular also at infinity ; it is 'd^" z d^-^ "^ ^ (fe™-2 +■-■+^"'11 where /j./j, ...,f^ are constants. The assignment of indices for s = determ- ines ^, ..., ^, and 80 determines the indices for 3 = co; and similarly the aesignment of indices for z = 'a determines those for 2 = 0. In fact, the indicial equation for ! = is p{p-l)...(p-W + l) = J^p(p-l).,.(p-™ + «-l)/„ and the indicial equation for : = (c is ( \TB{6-\-\) (fl-l- r=2( l)'"-"fl{fl + l) {fi+ K + 1)/" t t on e BY deot th fc the ts c n bo a a d 5 a r on fr ta 1 equat n m the form p+5=0 As re^rd the ntegrals, t ea y to vo fy n a cordance w th the general theory that the ntegral wh h belongs to a s mple r ot of the nd al ejuato for =0 a c n t t miltjlo of and that the ntegraL wl h bel n t j t i le x)t of tl t e iiiat o re tant mult pies of for a = 0, 1, ...,«-!. Es;. 5. Consider tlie equation i>ip = 3{l-3)w" + (l-2s)w'-iw = 0, wMch* clearly satisfies the conditions that its integrals should be regular, both in the vicinity of its singularities and for large values of :. To obtain the i utegrals in the vicinity of s=0, we substitute )w = eo2"-fc,i!' + i + ... + c„s" + ''-f..., and find zDiB = a„aH'', provided {a + «')^c„ = (o+«-^)^c„-i ; so that, writing f(. + ^){a + |)...(a + m-i)) = the value of lu is «> = c„s"(l+yi! + y22^ + ...). * It ia the differential ec[uatioa of the q^uarter-period in elliptie functions : fo detailed discussion of the equation, See Tannery, Ann. Ae Vka, Noi'm. Sup., S^r. S t. viii (1879), pp. 169—19*, and Fuohs, Crelle, t. lxsi (X870), pp. 121—136. y Google 130 EQUATION OF QUARTER-PERIOD [46. Tlie indieial equation is o*=0 : accordingly, the two integrals belong to the indes 0, and they are given by [a.. To particularise the integrals, we take c^^^^tt ; the first of the integrals then becomes .<„=i.{,.(iy..(H)'--..) say : and the second of them becomes Z (s), where -| 1 2 *" 2toJ say, where P™ 1 2^3 4'^"'"^2»i-i 2m' And now the two integrals in the vicinity of the origin are K(.}, L{f,). To obtain the integrals in the vicinity of ^=1, we substitute when the equation takes the form which is of the same form as in the vicinity of ^ = 0. Accordingly, the integrals in the vicinity of 2=1 are given by -TW, !■{')■ To obtain tbe integrala in the vicinity of 3 = aj, wo substitute 1 when the equation takes the form The indicial equation for (=0 is «("-i)+i-Oi and we find the equation for « to bo y Google 46.] IN ELLIPTIC rUNCTIONS 131 of the same form as in the tirat aad the second eases. Accordingly, tha integrals of the original equation in the vicinity of a — oo are The integrals are thus regular in the vi n ty of tl e t! ve nj, 1 r t cis 0, 1, to. Of these, the integrals K{z) i(, ) are s gmfl ant n the ioma n |e|<I, aay in D^; the integrals K{a!) L{x) ore s gi fic^nt n the domain. |a;| = |z-l|<I, say in iJi ; and the integiaU i*A if) ^L{t) aie 3 gn ficant in the domain \t\ < I, that is, \z\ > 1, aay in -0„. The ser es A ( ) d verges wi en 2= I, so that the integrals cease fco be significant for such a value. The domains D„ and Di have a common portion, so that values of z esist which are defined by |a <1, 1J-1|<L Within this common portion, the integrals K{z), L{z), K{a:\ L{x) are significant : so that, as K {£) aud L (s) make up a fundamental system, we K{^)=AK{z)^BL{z), L{:>^)=A'K{z)+B'L{^), where A, B, A', B' are constants. The values of the constants are determined as follows by Tannery. The integrals are compared for real values of a which are positive and slightly less than 1, so that, as s then approaches 1, K{e) tends to an infinite value. To obtain this infinite value, we note that, as by Wallis's theorem, wo have and therefore, for I'eal values of s between and I, we have The difference of the two quantities, between which the value of K{z) lies, is which increases as the real value of s increases and, for a = l, is that is, 1 - log 2, Kenofi wc may take .,(.)-5iog(i-.). y Google 132 EQUATIOK OF QUARTER -PERIOD [46. where ^^>.(3)>j7r-l + log2; find the values of z are real, positive, and less than 1. The result shows that K(s) is logarithmically infinite for s=l. Proceeding similarly with I{z) in the expression for L (s), we have, for real values of z between and 1, The difference of the two quaatitics, between which the value of iHa) s the real value of 3 increases and, for 3=1, is and therefore the foregoing difference is less than that is, less than (1 - log 2) log 2. Hence we may take where, for real positive values of z that are less than 1, 0<('W<(l-log2)log2. The expression can be further modified. Wo have S ^^"'<log3 I ^, for the values of z considered. The difference between these two series is I log 2-^^ the real value of s increases and, for 3=1, is S ^ (log 2 -£{„). 8 =-1 '-+. "^ 2m + l 2-111 + 2 1 y Google «.] IN ELLIPTIC FUNCTIONS aod therefore the difference is <.!,i^i)<^(l->'>l!^)' oil evaluating the ai sries. We may therefore take J^5^=J_^'^log2-("W where = -log(l-^)log3-^'(^), 0<t"(s)<aa-log2). Therefore, finally, w '0 have where so that i/(.}=-ilog(l-^)log2-.,(4 0<<i(s)<l-(log3)^; and the values of s considered are real, positive, and less tl In the region co mmon to i>„ and jD,, we have KIf)-AKif)+BL(i)-. and therefore, for real values of z less than (but nearly equal to) 1, that is, for real, positive, smaU values of x, ^(3^)^^*(a)-i^log^-2fllog^log2-4Cf,W+5{.(s)-ilog;i^}loge. When s tenda to the value 1, the term log^c log 2 tends to the value : more- over, K{!0) then tends to the value ^w ; hence, taking account of the infinite terms on the right-hand side, we have ^-1-45 log 2 = 0. Again, when % is real, small, and positive, m ia real, positive, and loss than (but nearly equal to) 1 ; hence 4-W..(»)^ilog(l-»).,(l-.)-ilogz, all the terms in which when l^l is small. {i-^)-iiog2=.iArw+sir(3)ioi finite except those i J+S/(s>, volving lo| a holomorphio function of s; thus A =1 log 2 ; o that A and B a y Google 134 EQUATION OF QUARTER-PERIOD [46. Similarly, for the other equation for values of x and s iu the ooramon region, wo have, for real, positive values of z less than I, that is, for rea], positive values of x that are small, ^Wlog»+/W-^'{,(.)-iIog(l-.))-4ir910E(l-,)log2 + ,,(.)l, hence, taking account of the logarithmically infinite terms on both sides, we ^' + 4S'log2 = -ff, Nest, tating the same equation for values of z that are small, real, and positive, 80 that x is real, positive, and less than 1, we have xir(.)+ff{i-wios.+/(,)).;rWiog «:+/«. When a; is nearly unity, jrM.,M-iiog(i-»), SO that K{3:)logx, for a; nearly equal to 1, is small : and it vanishes when a'isl. Also, for those values of ^, = -21og3log2-4.,(^); whence, equating coefficients, we have ijrff=-21og2. Thus 5'= --log 2, ^■=-(log2)=-w. Accordingly, when j lies within the portion common to the two domains Df^ and Z)j, defined by the relations irw.(iiog3)^(.)-ii(.) I iM.{"(iog2)>-,}irM-(i log 2) L(.) \ where x=l-~e. These results shew that, for complex values of j such that [zj^l, both X(i) and Z(2) converge. The fiist of them is a known result in the theory of elliptic integrals ; writing b = *^ ^-^^ Z(j)=£, AT (^) = Jf', we have an equation which is specially useful for small values of h Similarly, for Ta]u©a of i nearly equal to unity, we have y Google 46.J RIEMANN S P- FUNCTION l3o Ex. 6. With the notation of the preceding esample shew that, for values of z common to the domains /), and />„ as defined by the integrals Z'{a^), i (a'), t^KU), i*Z(;) are connected by the relations (Tannery.) E^. 7. Denoting the integrals of the equation in Es. S that are associated with the values 2 = 0, 1, cc. by ^, Z ; K', L' ; K", L" ; respectively ; denoting also the efiect upon a function Uof a, simple cycle round a point ahy \_U^, and of simple cycles round a and b in succession by [ U^r, , prove that [^li^-fs+f logaW'+^'i'; and express [L%, [L"],,, in terms of K', JJ. (Tannery.) Ex. 8. Discuss, in the same manner as in Ex, 5, the integrals of the (i) ^(l~.)W'-i^=0; (ii) 2(l-z)«7" + (l-j)«/ + J«.=0; (iii) 2(l-^)W' + w'-iw=0. RiEMANN'S P-FUNCl'ION. 47, It has already been proved {Ex. 3, § 46) that the only linear differential equation of any order other than the first, which is made completely determinate by the assignment of ita singu- larities and of the exponents to which the integrals belong in the respective vicinities of those singularities, is an equation of the second order which, if ifc have oo for a singularity, has two other singularities in the finite part of the plane. If the latter be at h,, k, then the transformation z—h_ho—b x~a z~k k c—a x~b gives a, 6, c in the ai-plane as the representatives of h, k; oo in the z-plane. The transformation manifestly does not aifect the order y Google 136 RIEMANN'S [47. of the equation, its sole result being to make a, b, c (but not now 00 ) singularities ; we shall therefore suppose this transformation made. Accordingly, we proceed to consider the properties of the function, which thus determines a differential equation ; they depend upon the properties initially assigned, which are taken as follows. In the vicinity of all values of s, except a, b, c (and not excepting od when a, b, c are finite), the function is a holooiorphic function of the variable. In the vicinity of any point (including the three points a, b, c), there are two distinct branches of the function ; and all branches of the function in the vicinity of any point are such that, between any three of them, a linear relation AT + A"P" + A'"P"' = exists, having constant coefficients A', A", A'". (So far as this condition affects the differential equation, it manifestly determines the order as equal to two.) As exponents are assigned to the three points, let them be a and a' for a : /3 and (3' for 6 : y and 7' for c ; these quantities being subject (§ 46, Ex. 2) 60 the condition a + a' + ^ + S' +y + y=l. It further is assumed that a — a.', ^ ~ ^, y — y ai'e not equal to integers. The branches distinct from one another in the respective vicinities are denoted by P„ and P„. ; Pg and Pp- ; P^ and Pyf. From the definition of the exponents to which they belong, the functions (s — a)-'P^ and (s - ra)-"'P„' are holomorphic in the domain of a and do not vanish when z — a. Similarly for b and c. After the earlier assumption, it follows that any branch existing in the vicinity of a can be expressed in a form where c„ and c^' are constants ; and likewise for branches in the vicinity of b and c. The assumption made as to a— a.', /3~0', 7 — 7' not being integers will, by the results obtained in §§ 35 — 38, secure the absence of logarithms from the integrals of the differential equation: it manifestly excludes the possibility of either of the branches P^ and P^', Pg and P^', Py and Py-, being absorbed into the other. y Google 47.] p-FUNCTioN 137 Biemann* denotes the function, which is thus defined, by y and the function itself is usually called Etemann's P-f unction. It is clear that a and a.' are interchangeable without affecting P; likewise yS and jS' ; likewise j and 7'. Also, the three vertical columns in the symbol can be interchanged among one another without affecting P ; six such interchanges are possible. Again, if P be multiplied! by (a: - afix-b)-^-' {x -c)', the effect is to give a new function, having a singularity at a with expo- nents a H- S, a' + 8 : a singularity at b with exponents — B — e, iS' — 8 — e ; and a singularity at c with exponents 7 + e, 7' + e. Every other point (including 00 ) is of the same character as for P. Hence / -a, w r<* ^ '^ ia:-hr 7 fS /3'- -e 7+6 Jr, -ey + e J the exponents on the right-hand side still satisfying the condition that the sum of the exponents shall be equal to unity, A homographie transformation of the independent variable can always be chosen so as to give any three assigned points «', b', c' as the representatives of a, b, c. Accordingly, let such a transformation be adopted as will make a and 0, b and <x> , c and 1, respectively correspond to one another : it manifestly is The indices are transferred to the critical points 0, 00 , 1 ; every other point is ordinary for the new function, as every other point was for the old. For brevity, the transformed function is denoted by ' Ges. IVerke, p. 63. t The sum o£ the indices in the factor a singularity for the new function. y Google where the two-term columns are to be associated with order. Also, sir ice — |5tH. it follows that, i ixcept as to a constant factor. (»-»)'<«-c)- («-!,)•+. '""' " *' "> agree ; and thus i, as regards general character, we have a!''{l~wyF 1" 1^ 7 A pC +S /5 -«-« T + W^'7 / U'+S ,S'-S-6 7' + As a — a', /3 — /3', 7 — 7' are the same for the P-function on the right-hand side as for the P-funetion on the left, Riemann denotes all functions of the type represented by the expression on the left by P («-«', 0-^', 7-7', ^'). In the transformation of the variable, the points a, b, c were made to be congruent with 0, » , 1 in the assigned order. A similar result would follow if they had been made congruent with 0, 00 , 1 in any order or, in other words, if 0, 00 , 1 be interchanged among themselves by horaographic substitution. As is known, six such substitutions a or, taking account of the association of the exponents with the first arrangement, the table of singularities, exponents, and variables for the six cases is Oool Owl Oool a 13 y x'; 7 /3 a 1-^'; /^ « 7 ^.i a' ^' i i ^' d ^' a' y' 7='^i--'; «7/3 ^3^ ; ^ "y " 1 _"^ ' y' a ^' a! 7' /3' /3' 7' a so that P-functions of these arguments with properly permuted exponents can be associated with one another. y Google 48,] i'-FUNCTIONS 139 48. The significance of the relation a + <x' + ^ + 0' + y + y' = l, in connection with the function, appears from the following con- siderations. When the singularities are taken at 0, x , 1, the axis of real variables, stretching from — co to + co , divides the plane into two parts in each of which every branch of the function is uniform; or, if the singularities be taken at a, b, c, then a circle through a, h, c divides the plane in the same way. In either ease, taking (say) the positive side of the axis or the inside of the circle, the linear relations among the branches of the function give P, =B,Fe + B,P^.] P. =(7.P^ + 0,Py) P„- = B,'P^ + B,Te. j ■ P,< = C'P^ + a^I\. J ' Bay P„, P„- = iB,,B, $Pp, P^.) = (i^P?, Ps'), I Pi', p; I p., p.- = ( 6', , C, 5P„ Py) = {cfP-„ Py) ; I o;, a; \ and with the usual notation of substitutions, lot Pg,P^, = (65P„, P,,), .Py,Py^ = (ciP.,P.,). Consider the effect upon any two branches, say P„ and P,', of circuits of the variable round the singularities. When it describes positively a circuit round a alone, they become e^'''°-P^ and ^'^' P^i respectively, so that, in the above notation, P„ P,; become ( e^-^, $P„, P.-). I 0,«-''| When it describes positively a circuit round h alone, then P^ and P^i become e'"''^Pp and e^'^'P^^ respectively ; and therefore P„ P„. become (fije^'^, J^jPa, P.')- I , e^'l Similarly, when it desciibes positively a circuit round c alone, P., P„- become (cfe'-^y, fc'^^P., P.'). yGoosle 140 RIEMANN'S i'-FUNCTION [48. Accordingly, when z describes a simple circuit round a, b, c, tho initial branches P^, Pa' are transfonnod into branches (oj.-., Vcme-O. jsf,,-, JP..P..), , 6''"T'[ I , (f"'f'| I , e=°'"' s.y nP.. P..). Such a circuit encloses all the singularities of the functions ; and therefore* each of the functions returns to its initial value at the end of the circuit, so that (7)=(1, 0). |o. l| The determinant of the right-hand side is unity; hence the determinant of / is unity, and it is the product of the determinants of all the component substitutions. Now as (c) and (c)"' are inverse, the product of their determinants is unity ; and likewise, the product of the determinants of (b) and {b)~^ is unity. Hence we must have an equation which is satisfied in virtue of the relation a+a' ^ + (i' + y + y'-^l: the sum of the exponents could be equal to any integer merely so far as the preceding considerations are concerned. In the present instance, the property, that a function returns to its initial value after the description of a circuit enclosing all its singularities, can be used in the form that the effect of a positive circuit round c is the same as the effect of a negative circuit round a and round b. Applying this to P^, we have C,Py^^ + (?,Pye^'" = e-="" (^i^pe-^" + BJ^^.e-"^'^) ; and i'rom the expressions for P„, we have C\P, + C,Py = B,Pff + B,Pp.. As Pp and P^' are linearly independent of one another, it follows that e*i™*— eV"* must not be zero, that is, 7 — 7' must not be an integer. Similarly for a — a' and 0- &■ Ex. Prove, by means of those relations, that O,'" S/8in(a'-(-^+y')ir Bj'sin(a'+^'+y),r ' £= >-')-*_ A«i°('' + g + y)T _. ga'^iti(n + g' + v)^ G4 £,'8m(a' + jy + v)n- iJ2'sin(a'4-(3' + r)7r" (Eiemann.) * T. F., % yo. y Google 49.] determines a differential equation 141 Differential Equation determined by Eiemann's p-function. 49. As regards the differential equation, associated with these P-functions, and determined by the assignment of the three singularities a, b, c, and their exponents, we know that it must be of the form d\v A'z-' + B'z + C d^'^{,~a){z-b){z-c)d^^ {z-af{z-bf(z-cr which (§ 46) secures that a, b, c, x are points in whose vicinity the integrals are regular. Now the singularities are to be merely the three points a, b, c, so that oo must be an ordinary point of the integral. Taking the most general case, when the value of every integral is not necessarily zero for s = x , we have an integral where Kt, does not vanish. Substituting, we havu the unexpressed terms being lower powers of 2; hence (2 - A-) K, + A"K, + \B" + 2A" (a + 6 + c)] K„ = 0, and so on. Using the result that A" = 0, the equation may 1 written in the form d'-w f A B G\ dw U^'^\T-a^~z-b^z-c}~dz Forming the indicial equations for the singularities, we have y Google 142 DIFFERENTIAL RQUATION DETERMINED [49. as the indicial equation for a ; and therefore, as its roots are to be a and a', it follows that A = 1 —a— a', X= aa' {a—b)(a — c). Similarly 5 = 1-/3-/3', fi = 00'ib-a)(b-c). C =1 —y— y, V — 77' (c —a)(c — b). Moreover A'=A+B + G=% on account of the value of the sum of the six exponents; tlio condition is thus satisfied by B"= 0. All the quantities are thus determined, and the equation has the form* rf'w n-a-a' l-fi-^'l-j-y'\dw dz^ \ z-a z—h z-c ) dz [ <.a'{a-h){a-c) &0 (b-a){b -o) 77' (c - a) {c - 6)1 from the mode of construction we know that the integrals are regular in the vicinity of the singularities a, b, c, and are holo- morphic for large values of s. This is the differential equation, vith (and determined by) the function (a b c W /3' 1 The branches of the integral in the vicinity of a are P, , P„-; those in the vicinity of b are Pf,, P^i ; and those in the vicinity of c are P,, P,-. Passing to the ibrm of the function represented by U' /3' 7' ' where the three singularities are 0, =0 , 1, we deduce the associated differential equation from the preceding case by taking c( = 0, &=«, c = l; • Firat given by Papperitz, Math. Ann., t. xiv (1865), p. 213. y Google id.] BY RIEMAKN'g P-PUKCTION 143 after a slight reduction, the equation is found to be d'w l-ii-a'-(l+i3 + /3');; <i» ((«■"'' 2(1-2) i , W-(aa' + gff'-T7')2 + gi3V „ + 2-(l-2)- The branches of the integral in the vicinity of the origin are P,, P^', so that ^""'Po, z~°''P^' are holomorphic functions of s, not vanishing when 2 = 0; those in the vicinity of a = 1 are Py, Py, so that {z — l)'"'Py, {s — l)~*'Py' are holomorphic functions of z—l, not vanishing when 2 = 1; and those in the vicinity of s — <rj are Pg, Pp., so that z^P^, s^'Pp-are holomorphic functions of — , not vanishing when z=<r^ . Lastly, passing to the form of the functions included in P(t.-«'. /S-/?-, 7-7. »). we saw that they arise from the association of arbitrary powers of s and 1 — 2 with the above function in the form and that they lead to a function /. +S, ft-S-,, y+. '. -^W + S, ^'-B-,, 7' + .'/- Thus we can make any (the same) change on a and a' and, as they are interchangeable, we can select either for the determinate change; accordingly, we take say, as the modified exponents. Similarly, we can make any (the same) change on 7 and j : we take ry —y = 0, J — y = V — ^ — /i, say. Then the new values of the exponents for co are /3 + a + 7, -\ say, and yGoosle 144 INTEGRALS OF THE EQUATION OF [49. on reduction : or the exponents are 0, 1 - /' , for 2 = , \, /t , for s = CO , 0, v~\-fi, for s = 1 . Their sum clearly is unity : moreover, with the preceding hypo- theses, the quantities 1-c, /^ —X, v -\ — fi are not integers. Specialising the last form of the equation by substituting this set of values for a, a', &, ^', 7, 7', we find the equation, after reduction, to he which is the differential equation of Gauss's hypergeometric series with elements X, /t, p. Either from the original form of the P-function, or from the resulting form of the equation, the quantities X and /j. are interchangeable. 50. Taking the equation in the more familiar notation , dhu , , ^ -, , 1 dw -, „ so that tho exponents are 0, 1— 7, for z^O; a, 0, for 3=x; 0, 7 ~ a — (8, for ^ = 1, we use the preceding method to deduce the well-known set of 24 integrals. Denoting as usual by F{a:, 13, 7, z) the integral which belongs to the exponent zero for the vicinity of z = 0, we have «{« + l)M8 + l) 1.2.t(t+1) assignitig to the integral the value um'ty when z= 0. If z'(i-^yF{cL',^;y',z) be also an integral, then the exponents for each of the critical points must be the same as above ; hence S, B + l-y =0, 1-7 , for 3 = , e, e + j'-a'-^'^O, 7-a-/3, for 2 = 1 , a'-S-e, (3'-S-e =«, /3 , for s = oo. Apparently there are eight solutions of these equations ; but as a and y3 can be interchanged, and likewise a' and 0', there are only four independent solutions. These are : — y Google 50.] THE HYPERGEOMETEIC SERIES 145 I. S = 0, e = ; giving a' = a, ^' = /3, 7' = 7 ; and the integi'al is II. S = l-7, e = 0; giving a' = l + «-7, ^' = 1+^-^, 7' = 2 — 7 ; and the integral is 2'-rf (H-a-7, 1+^-7, 2-7, ^); III. S = 0, e = 7 — a — /3; giving a' = 7 — n, /3' = 7 — /3, 7' = 7 ; and the integral is (l_^)v— fljr(^_a, 7-A 7, ^); IV. S=l-7, e = 7-a-^; giving a' = 1 - /3, ^' = 1 - a, 7' = 2 — 7 ; on interchanging the first two elements, the integral is ^'T (1 - z)y---f F(l - «, 1 - A 2 ~ 7, z). Next, it has been seen (§ 47) that, in the most general case, P-functions can be associated with a given P-function, when the argument of the latter is submitted to any of the six homographic substitutions which interchange 0, 1, x amoDg one another, provided there is the corresponding interchange of exponents. Taking the substitution e'z = 1, the new arrangement of exponents a, , for / = 0, 0, 7-a-/?, for s' = l, 0, 1-7 , for /=co; heirce, if 2"(l-«')--f(«', /3',7'-») is an integral, we must have S, S + 1 - 7' - 0, ,3 for /-O, e, e+7--ci'-/3'-0, i-a-fi, for s'-l, a--8-e, /3'-S-. -0, 1-7 , fop /-«. Again tlrere are four independent solutions ; they are i — IX. S-a, «-(); givingf'-a,/3'-l+«- and the integral is - 7, 7' - 1 + « - -13 z-fU 1 + 11-7, i+«-A lY y Google 146 rummer's [50, X. g = Ae = 0; giving «' = y3,/3' = l+;3-x 7=1 -« + /3; and the integral is z-^F^^, 1+^-7, l-a+/3, J); XI. B = ^, e = j-a-0; giving £('=7- a, ^' = 1 - a, 7' = 1 - a + /3 ; on interchanging a' and ff, the integral is •(-r"-(' XII, S-a, e = 7-a-^; giving a'^^-^, /3'=l-/3, 7' = 1 + a — ^ ; on interchanging a' and ^', the integral is s-'(l-^-J~^~^ f(i-^, 7-/3, l+«-A J). The remaining four sets, each containing four integrals, and belonging to the substitutions respectively, can be obtained in a similar manner*. Tliey are ; — V. f(o, ft a + ,3-7+1, f); VI. (l-f)>^F<«-7+l,/3-7 + l,<. + /3-7 + l, 0; VII. fv--«Ji'(7-ii, 7-ft 7-a-3 + l, 0; VIII. (1 - f)-» f— « ^(l-a, l-ft 7-«-^ + l, f); in which set £; denotes I —n: XIII. fj?(«, 7-A «-;3+l, 0; XIV. CF{/i, 7-«, /3-« + l. 0; XV. (l-i;)"+>?-*'(a-7+l, 1-13, 1-13 + 1, (); XVI. (l-t)-"»f'.f'(|3-7+l. 1-ci, /3-1I + 1, a; in which set if denotes .j : SVII. (l-f)-i?(», 7-ft 7, f); XVIII. (l-flBfCft 7-a, 7, 0; * The complete set ot expreasions, differently obtained and originally due to Knmmer, ate given in my Treatise on Differential EquaUom, (2nd ei.), pp. 192— 194; the Eoman numbers, used above to specify the cases, are in accord with the numbere there used. y Google 50.] INTEGRALS 147 XIX. f-^(i-f)'J?(a-7 + i, :-A 2-,, O; XX. i"'{l--i;}'F{/3-i + l. l^c. 2-y. 0; in which set f denotes ; and s — 1 XXI. (l-f)"*'(«. <"-7 + I. « + ^-7 + l. E)i XXn. (l-f)»F(A^-7 + l, a + /3-7 + l, t); xxm. cy'-'0--tyF(i-a, ,,-a, i-a-^ + i. Oi XXIV. tr-'(i-O'r(i-i3.j-0. j~„-i3 + i, 0; in which set Jf denotes . The preceding investigations have been based upon the assumption, among others, that no one of the quantities is an integer or aero : the determination ot the integrals of the differential equation when the assumption is not justified, can be effected by the methods of §§ 36—38. Consider, in particular, the int^rals in the vicinity of s = 0, when l~--y is an integer ; there are three cases, according as the int^er is zero, positive, or negative. We substitute ».cy + «, .'+' + . ..+c..'+- + ... in the equation ; and we find zBto^e{e+y-l)c„/, provided (^-l +„ + fl)...(n + g)(w-l + g + d)...(g + fl) ''" (>i + 6) {l + e){n. 1+7+5) (7 + -^) "■ (i) Let 1—7 = 0, so that the indiciil equation ls S^ = : then the two integrals belong to the mdcs 0, and one of them certainly involves a logarithm; and they aie t,nen by «-• m.. The former, when we take Cq—I, is F(a, ft 1, z), with the usual notation for the hypoi^eometric function ; as the coefficients ai'e required for the other integral, we write F{a,0, 1, s) = l+Ki2+«52^+... + -=,.j"+.... y Google 148 THE HYPEIIGEOMETRIC [50. The second integral, when to it we add C(, again being made ec[ual to unity, becomes where ij/ (m) denotea ^ {log n (m)). (ii) Let I — y be a positive integer, say p, where p>0. The indicial equation, being S(S—p)=0, has its root-s equal top, 0. We have '^^ (n+e)...ii+6)(n-p+s)...(i-p+e) '■ Of the two int^r .Is, that, which belongs to the greater of the two exponents, is equal to z''F{a-i-p,^+p, i+p,z}, when we take <;|,=0. The other integral may or may not involve logarithms. If it is not to involve logarithms, then, as in § 41, the numerator of o^ must vanish when 8=0, so that (p-l+a)...aip-l + 0)...0 must vanish ; in other words, either a or 3 must be zero or a negative integer not less than y. When this condition is satisfied, the integral belonging to the index zero is e&ectively a polynomial in z of degree — a or - /3 aa the case may be, and it contains a term independent of i. When the preceding condition is not satisfled, the integral certainly involves logarithms. In accordance with § 36, we take Ga=ce, so that w = (7 S *■■ , (^-l + a + g)...(a + fl) (»-l +fl+g)...(g + g ) {n + 0)...{l + e) (n-p + e)...{l-p + 0)''' There are two integrals givOTi by M- [Sh The first is easily seen to be a constant multiple of z''F{a+p, 0+p, l+p, s), thus in effect providing no new integral. The second, after redm mating C^l, is J-> (»-l+.)....(«-H-|3)...3 + „.»!(~-y)(«-l-rt...(l-j.)' +(_i,.-i; (— n-.).-.(»-n-w-.g ,. y Google 50.] =^(^_H.„)+^(«_i+3)->;,(«)-f(»^^). (iii) Let 1— 7 be a negative integer, say -q, where 5>0. The indicial equation, being ${S-[-q) =0, has its roots equal to 0, - q. We have («-l+„+g)...(„+g)fa-l+g+g)...(3+g) (™+6I),..(l+d){«+j+(9)... (1+3+19) ^' The greater of the two exponents is 0; the integral which belongs to it, on making Cf, = \, becomes F{a,»,l-¥q,z). Tbfs integral which belongs to the exponent — g may, or may not, involve logarithms. If it is not to involve logarithms, then, as before, the numerator of tfj must vanish when ^ = - 5, so that (.-l)...(a-,;)((i-l)...(e-,) must vanish : hence either n or j9 must be a jxisitive integer greater than and leas than y( = l + q). When the condition is satisfied, the jnt^ral is a polynomial in s~\ beginning with £~', and ending with a~" or 3~P, as the case may ba. When the preceding condition is not satisfied, the integral certainly involves logarithms. As before, in accordance with § 36, we take e,~(»+})Jr, ... „. («~i+.H-»)...(»+») ( «-i+e+<)-0-K ),„i ,,,«H-. Two integrals are given by The first is easily seen to be a constant multiple of /■(n, 3,1+ J, 4 so that no new integral is thus provided. The second, after reduction, and making K= 1, is +(-1,.-. J (»-lt -g ).-(°-g)('-l+g-g)-(8-?) ,,— , *„=-K«-l+«-?)+^(7i-I+e-?)-v('(»i)-i|'(n-s). The integrals are thus obtained in all the cases, when y is an integer. y Google 150 EQUATIONS OF THE SECOND ORDER [50. Siroilai' treatment can be applied to the integrala of the equ^ition, when ■y-a— 3 i^ *" integer, positive, zero, or negative, contrary tg the original hypothesis as to the exponents for 3=1 ; likewise, when a-,8 is an integer, positive, zero, or negative, contrary to the original hypothesis as to the exponents for 3=a>. These instances are left as exercises. Note. There is a great amount of literature dealing with the hypergeometric series, with the linear equation which it satisfies, and with the integrals of that equation. The detailed properties of the series and all the associated series are of great importance : bub as they are developed, they soon pass beyond the range of illustrating the general theory of linear differential equations, and become the special properties of the particular function. Accord- ingly, such properties will not here be discussed: they will be found in Klein's lectures JJeher die hypergeometrische Function (Gottingen, 1894), where many references to original authorities will be found. Equations of the Second Order and Fuchsian Type. 51. No equations of the Fuchsian type, other than those already discussed, are made completely determinate merely by the assignment of the singularities and their exponents. It is expedient to consider one or two instances of equations, which shall indicate how far they contain arbitrary elements after singu- larities and exponents are assigned. Suppose that an equation of the second order has p singulari- ties in the finite part of the plane and has co for a singularity ; the sum of the exponents which belong to these p + 1 singularities is (by Ex. 2, § i6) equal to p — 1. Now let a homogiaphic substi- tution be applied to the independent variable and let it be chosen so that all the points, congruent to the p+1 singularities, lie in the finite part of the plane. Thus « is not a smgulaiity of the transformed equation: there are p+1, say re, singulaiities in the finite part of the plane: and the adopted transfoimation has not affected the exponents, which accordingly aie transferied to the respective congruent points. Hence, when an equation of the second order and Fuchsian type has n singularities m the finite part of the plane and when infinity is not a singularity, the sum of the exponents belonging to the n points is equal to n — 2. For y Google 51.] AND FUCHSIAN TYPE 151 such an equation of the second order, let the singularities and their exponents be then Let ^-.VW = (»-<..)(2-a.).. .<«-«.); then, as the equation is of the second order and as all its integrals are regular, it is of the form where F-^ and F^ are polynomials in z of orders not higher than n — \ and 2n — 2 respectively. Also, let F _ A, A^ A„ . ■^ s—a, s — a^ '" s — a„ ' and let F^ = F^(s) - A"e'^-^ + B"2f^-' + CV-' + .... The indicial equation for the point e = a,, is '(o-'^ + ^-' + WmT"'' and therefore a, + S, = l~A„ X A,. = n -%(«, + 0,) — '2, and therefore the polynomial Fi is of the form F, = 2s""'' + lower powers of z. Again, X is to be an ordinary point of an integral ; hence, talcing the most general case, we must have an integral y Google 152 EQUATIONS OF [51. where K,, is not zero ; for otherwise we should have a special limitation that every integral is zero at infinity. Substituting, so as to have the equation identically satisfied, and writing (so that Sa = 2), we find, as the necessary conditions, = K,A", = (2 - s,) K, + K,A" + K^ \b" + 2^" 2 a\ , <i = {'l> -2s,-\- A")K., + kJ- s, + B" -^lA" i re,) + K„ \a"U I «,' + 2 2 ara)j + 2S" I a, + C'\ , and so on. The first gives A" = 0; then the second gives both of these equations leaving Kg and Ki arbitrary. The third equation then gives and so on, in succession. The remaining coefficients K are uniquely determinate; they are linear in Ki and K^, the various coefficients involving the singularities and their exponents, as well as the coefficients in F^. The equation therefore has z = <x,for an ordinary point of its integrals, provided F^is of order not higher than 2n — 4. The equation can, in this case, be expressed in a different form. Let ^= = 1(0"^-* + ...) = Pn--4 + + ^ +,.. + — ^^ , s-a, z-as z-a^ where P,^^ is a polynomial of order n — i. (Of course, if 2n ~ 4 is less than n, which is the case when n = 3, there is no such polynomial.) As the coefficients in F^ are not subject to any further conditions in connection with the nature of 3= co for the y Google 51.] FUCHSIAN TYPE 153 integrals, any values or relations imposed upon \,, X^, .... >.„ and the coefficients in Pn-t must be associated with the singularities. The equation now is ^„ /| l-»,-ffA , 1/ |_X, A _„^ The indicial equation for a = a, is ^ (fl - 1) + (1 - a, - ^.) ^ + ^,''^^— ^ = 0, and its roots must be ctr, 0r- thus and therefore the equation is It follows that the only coefficients which remain arbitrary are the ji — 3 coefRcienta in the polynomial P,_ (where n ^ 4). When the polynomial P„_i is arbitrarily taken, the foregoing is the most general form of equation of the second order and of Fuchsian type, which has n assigned singularities in the finite part of the plane with assigned exponents, and has oo for an ordinary point of its integrals. This is the form adopted by Klein*. If a new dependent variable y be introduced, defined by the relation w = y {z — a^Y' {z - a^'"' ... (z-a^)'^, then the exponents to which y belongs in the vicinity of a^ are the difference of which is the same as for w ; but s = co will have become a singularity, unless Pi + Pi + ...+p^>0. Now ^1 {(<.. ~ p.) + (0. - p,)) = n ~ 2 - 2 1^ />, ; and therefore i_ !1 - (», - p,) - (A - p,)i = 2 + 2 l^p,. * VorlemngenUberlineareDifferentialgleichungendei-ziueilenOrdnanglGottmgeu, 1894), p, 7. y Google 154 EQUATIONS OF [51. Hence, if 3= ic is not to be a siDgularity, the quantities p,, ..., p^ cannot all be chosen so that each of the magnitudes vanishes. Conversely, if the quantities p^ be chosen so that each of these magnitudes vanishes, then z = oo has become a singularity of the equation ; having regard to the form of w for large values of 2, we see that and 1 are the exponents to which y belongs for large values of z ; and the differential equation for y is easily seen to be where P„_3 is a polynomial of order n~S. This equation, however, has n singularities in the finite part of the plane, and a specially limited singularity at s = co : we proceed, in the next paragraph, to the more genera! case. Note. The indicia! equation for a = oo in the case of the equation for w is 0(0 + l)-0j^(l-a.-^.) = O, that is. The root = gives an integral of the form and the root = 1 gives an integral of the form ^(;-*: both of which are holomorphic for large values of |s[, so that all integrals are holomorphic functions of - for large values of \z\. In this case, oo is not a singularity of the integrals : it can be regarded as an apparent singularity of the differential equation, and (if we please) we may consider and — 1 as its exponents. £x. Shew that the preceding equation can be eshibited in the form <!.^^^)-t:.< y Google 51.] FUCHSIAN TYPE 155 where the n constanta c,, ..., o^ satisfy the three relations 2 c, = 0, S C^r+ 2 c^0r = O, 2 c^/ + 2 2 Orl3^ay = 0, and otherwise are arbitrary in the most general case. (Klein.) 52. Now consider the equation of the second order and of Fuchsian type, which has m singularities in the finite part of the plane, say Oi, a^, ..., (X«, with exponents a, and /9i, ..., a„ and ^n, respectively, and for which m also is a singularity with exponents a and /3 : the exponents being subject to the relation Let il' denote (z — a^)(s-a^)...(s — an): then the equation is of the form w"+( I -^"l w' + -^^ w = 0, where (? is a polynomial of order not higher than 2n - 2. When G is divided by yjr, we have a polynomial of order n-2 and a fractional part : and so we may write The indicial equatioH for 3 = a^ now is 4,= l-a, -ft, holding for r = 1, 2, . . . , jj. The indicial equation for i -l)-0 2 ^, + A„_, = 0, -X Ar-1, /(„_ = a^; the former being satisfied on account of the relation between the exponents. The equation thus is w" + 2 - .+;..+ s?^^!±>^>l-o, y Google 156 A NORMAL FORM OF EQUATIONS [52. the coefficients ht,, h^, ..., ^^,-3 being independent of the singu- larities and their exponents. When a new dependent variable y is defined by the trans- formation w = (2 -aO"" (^ -«.)"»... (^ - «„)-"y, then the exponents ofy for a, are and ^, — a,, say and X^, this holding for r=l, 2, ,,., n: and its exponents for x are BL+% a,, 0+i a,,, = <r, T say : where a- + T+ X Xr = ii- 1. The function y is, in general character, similar to w: it has the same singularities as w, and it is regular in the vicinity of each of them but with altered exponents : and it thus satisfies an equa- tion of the second order and Fuchsian type, which (after the earlier investigation) is where i„_3, ..., k„ are independent of the singularities and their exponents *. This transformation of an equation to an equation „ , G.-, , ^ 0.-. „ where Fa-i, F^_^, G^-i, G„^2 are polynomials of order indicated by their subscript index, appears to have been given first by Fuchsf, The simplest example of importance occurs for n = 2, when the hypergeometric equation is once more obtained. 53. It is well known that, when y is determined by the equation y" + Py'+Q = 0, * The equation for y can be obtained by the direct substitution of the expreafiion for w in tlie eailier differential equation for lu. When reduction takes plaee, there t Heffter, Einleitung in die Tkeorie der Unearen DijferentialgUichungen, (1894), p. 23i. y Google 63.] OF FUCHSIAN TYPE 157 and a new variable F is introdueed by tbo relation the differential equation for ¥ is where i.e-if-jp.. In the case of the preceding equation, the relation between y and Y is SO that F is a regular integral in the vicinity of all the singular- ities and of t» , the exponents being i(l-Xr), ^(1+Xr). fo5- s = .a„ ('■ = 1. ■-.™). and ■^ (- 1 + o- - t), ^ (- 1 - o- + t), for s = iO . From the form of P and Q, it is easy to see that I-i^^ = polynomial of order 2w — 2 ^ L ,.i^-»J where P„_2 is a polynomial of order )i — 2, say P„ - Bz— + (,_, 2" + . . . + i,. In order that ^ (1 — V), ^ (1 + V) niay be the exponents of a^ for the equation Y" + IT = (1, they must be the roots of e(e-V) + -,?'--0: hence JJ,,.1(1-V)^''(«,). In order that ^ (— 1 + a- — t), ^ (— 1 — ct + t) may be the exponents of <Xi for the same differential equation, they must be the roots of 0(<f. + l) + C = O: hence C-ill-(.7^T)-l. y Google 158 KLEIN'S NORMAL FORM [53. The remaining' constants l^, l^, ---, Ins ^^^ expressible as homo- geneous linear functions of &„, i,, .,., k^^i, so that they are inde- pendent of the singularities and the exponents : and thus the equation is -fJ+|[lll-(--)1-" + !.-.^"+...+i. r)].o. Corollary. For the original equation, x was a singularity of the integrals with exponents a- and t. If it were only an apparent singularity of the original equation, so that the integrals are regular for laj-ge values of \s\, then we have the ease indicated in the Note, § 51, so that we can take <T, T = 0, -1. The modified equation now is For this differential equation and its integrals, the exponents to which the integrals belong in the vicinity of Or are ^(1 — X,), J(l+Xr); but 30 is now a singularity of the integrals, and the exponents for a = co are 0, — 1, so that s = i» is a simple zero of one of the linearly independent integrals of the modified equation. dY These forms of the equation, from which tho term in -,- is absent, are the normal forms used by Klein. The simpleBt example of the class of equations, not made entirely determ- inate by the assignment otthe singularities and their exponents, occurs when there are three singularities in the finite part of the plane and oo also is a singularity. By a homc^raphio transformation of the variahle, two of the singularities can he made to occur at and 1, and cc can be left unaltered ; let a denote the remaining singularity. Let the exponents he 0, 1-X(,forz = 0; 0, l-Aifori=l; 0, X for j = u ; o-,rfor2=tt>; where cr-|-T-X„-Xi + X = 0. Then the differential equation is y Google 53.] EXAMPLES 159 where q is the (sole) arbitrary constant, left undetermined by the assigned properties. The integral of this equation, which is regular in the vioinitj of 3=0 and belongs to the index 0, is denoted* by F{a, q; cr, t, Xj,, X,; s). If «•=!, 3=1, the equation degenerates into that of a Gauss's hypergeometrie series ; likewise if a = 0, j=0. Ex. 1. Verify that, when a = ^, the group of substitutions , i_, t ?ri !zi ^ ifci) i!_ '■ •■ i- , • .-V .-!• .-1 ' .-i' interchanges among themselves the four points 0, ^, 1, co . Prove that, when a=~\ and when a =2, there is in each case a corre- sponding group of eigbt substitutions interchanging the points 0, 1, a, cd among themselves: and that, when a=^{\+i^2) and when a=^{\-i^3), there is in each case a corresponding group of twelve substitutions. Construct these groups. (Heun.) Ex. 2. Prove that there are eight integrals of Heun's equation of tte i^{z-\f{z-a)yF{a,q; ,/, r', V. V 5 4 which are regular in the vicinity of the origin and have the same exponents as F{a, q; a; T, X„, X, ; ?). Hence construct a set of 64 integrals for the equation when «=J, which correspond to Kummer's set of 24 integrals for the hypei^ometric series. Indicate the corresponding results when a=-\, 3, ^(1 + W3), ja-i^a)- Ex. 3. A homogeneouiS linear differential equation of order n is to have n singularities ts,, Qj, ,,., a^ in the finite part of the plane and also to have 00 for a singularity : the integrals are to be regular in the vicinity of each of the singularities, and the exponents of 01^ are to be 0, 1, ,.., n — 2, a,, (for c=l, ..., «), while the exponents of ro are to be 0, 1, ..., ra— 2, a, so that a + J^«, = («-l)^ Shew that the differential equation is where ij» (s) = (s - Oj) (s - %). , .(j - o,), the coefiicient Eg (b) is a polynomial in z of order not greater than », (for 3=1, ..., n), and .E,(2)=£(a.-«+l)^-^. (Pochh amm er. ) ' Heun, Math. Ann., t. xxxiii (18891, PP. 161—179, who has developed some of the properties of these equations, and has applied them, in another memoir fl.r,., pp. 180—196), to Lajnf s functions. y Google 160 equations op fuchsian type [54. Equations in Mathematical Physics and Equations of FucHsiAN Type. 54. These equations of Fuchsian type include many of the differential equations of the second order that occur in mathe- matical physics; somebimes such an equation is explicitly of Fuchsian type, sometimes it is a limiting form of an equation of Fuchsian type. One such example has already been indicated, in Legendre's differential equation (Ex. 1, § 46). Another rises from a transform- ation of Lame's differential equation which (| 148) is of the form 1 d^ . , , 71 « -5J,+^S'W + -B = (l. whore A and B ai'O constants*. Writing (•(,). c,, so that ic is a new independent variable, we liave d% / i , J i \diD , , Ax + B „ ejdx ' {X The singularities of this equation are Bi, e^, e^, oo; the exponents to which the integrals belong in the vicinity of Si, e^, e^ are and J, in each case ; the exponents, to which they belong for large values of a;, are the roots of the equation The new equation is of Fuchsian type : and, in this form, it is frequently called Lamp's equation. An equation, similar to Lamp's equation, but having n singu- larities in the finite part of the plane, each of them with and J as their exponents, as well as ^ = oo with exponents a and 0, such that (§ 52) «-^^:=i«~l, is sometimes called Lame's generalised equation. By § 52, it is of the form w" + w' i -J--I- „'^"^° w = 0, ,^^z a^ Ji(^-a,) " This 13 the general iorni; tho -value ~fi(K + l) is assigned (i.e.) to A, in order to have those eases of the general form which possesa a uniform integral. y Google 5*.] IN MATHEMATICAL PHYSICS 161 where Gn-i is a polynomial of order n~2, the highest term in which is a^s"~^. 55. The equation of Fuchsian type which, nest after the equation determined by Riemann's P-functioii, appears to be of moat interest is that for which there are five singularities in the finite part of the plane, while ^ = x is an ordinary point. The interest is caused by a theorem*, due to EScher, to the effect that when the five points are made to coalesce in all possible ways, each limiting form of the equation contains, or is equivalent to, one of the linear equations of mathematical physics. Let the points be a-^, a^, (h, Wj, <h, with indices a^ and ^r, for r=l, 2, 3,4, 5; then and the equation (p. 153) is where -^ = 11 (z — Or), and Pi is a linear polynomial Ax + B. The substantially distinct modes of coalescence are :— (i), ttj and ttj into one point ; (ii), Ma and a^ into one point, a, and a,^ into another ; (iii), tta, 0.1, 1X5 into one point ; (iv), Oj and a^ into one point, ctj, at, Kj into another; (v), at, Ob, ai, a, into one point ; (vi), all five into one point ; and the various cases will be considered in turn. Gase (i). Let the indices for a^, a^, a^ be made 0, ^ for each point ; then, as -i/^' (0.4) = 0, 1^' (dj) = in the present case, and * Ueber die Eeihenentwiokeluagen i!er Potentialthoorie, GStt. geirSnte Preis- sehri/t, (1891), p. 44; and a sepaiate took undpr the same title, p. 193. Sbb also Klein, Varlesungtii iiber Uiieare DiJ/ircnhiljl itliuniie?i dee xxaeiten Ordmiiig, (1894), y Google 162 bocher's theorem on [55. the equation is * i F Write 2 — »! = - , (ttr — «4) ^r = !> foi' '■ = 1, 2,3; the equation becomes d?wdw( i , \ . \ \ , C^ + I> ,.. ^ in effect, the preceding ungeneralised Lamp's equation. Case (ii). The equation becomes ^.,^ ^,|l -a.-ft ^ l^K-^8 ' ^ l-.-'-ri +(- .-».)(.-';).(.-,.). {^-"^/--''-'''--'4°''- itfber coalescence of the points, where 1-a' -jS' =2-«,-/3,-«,-/3„ 1 - a" - y9" = 2 - a, - ^, - a, - ,3, , and therefore fli + /3i + a' + ^' + a" + /3" = 1. Writing = (z — a^{z — a.^{z — a,^, we have the coefficient of -^ in the form z-a, (z- a^) (z - ft.) ' where Q,, like P„ is an arbitrary linear polynomial. Thus Q, contains two arbitrary coefficients ; these can be determined so that {z — a-t) {z — at) z — o^ ^ - «j ' and then the equation becomes .,/■ ^ .„■ | i-''.-ft J. i-«'-g ^ i-»"-ri «1 s-o, J-o, a-o, J y Google 55.] EQUATIONS OF FUCH8IAN TYPE 163 Owing to the form of $ and the relation 2 (a + (S) = 1, this is the equation of Eiemann's P- function (§ 49). When we write ai = l; (1^ = — 1; ai = <xi; a„^, = 0, 0; a', ^' = 0,0; a", ^" = ~n,n+l ; the equation becomes that is, {1 - ^)w" - 2zw' + n(n + l)w = 0, which is Legendre's equation. Case (iii). Let a, , A = 0, i ; a, , A = 0, i ; so that 1-«,~A + 1 -04-^ + 1 -«>-ft = l. After the coalescence of the points, the equation is .„"..„.r_i_^.J- -a,l (^-o,)(F -»,)(« -,..)■• where P) is a linear polynomial, say {^ {z — as) + B}{a, — a^)(as — Oi). Now let after some easy reduction, the equation becomes — , dta f 1 1 Us — Oj flj — Hj] 1 Let 01 = 00, (Xj — tt2 = — 1: the equation is d'w , 2ai — 1 (^ A + Writing a) = sin'' t, we have ^-^w{A + B sin= = 0, yGoosle 164 LIMITING FORMS OF AN [55. which is known* as the equation of tho elliptic cylinder. This equation will be discussed hereafter {^ 138 — 140). Case (iv). Let a„ A= 0, i ; '^.03=0, ^; so that, as in the 1-0,-/3, + ! -0,-^4+1 -a,-^, = l. After coalescence of the points, the equation is P, - (hf' Let s-«, = i P, = |«(^-«,) + /3}(a,-«,)^ c{a,-a,) = l; then the equation becomes dHv 1 f^w a + 00) or, taking we have ci% 1 dw dhu 1 dw /4a , ^\ ay" y dy \y^ J which includes Bessel's equation, sometimes called the equation of the circular cjKnder. Case (v). Let «i, A = ", ^ ; then 2^(l-«,-^,) = |. and the equation, after coalescence of the points, becomes «." + a.'(^- + -L~) + ^ ?1 ^^„ = 0. Let 2-„, = i, P, = |»(2-o,) + ,31(a,-o,). 6<»,-o=)-l; then the equation is dhu dw ^ a + ^x ^ da^ aw x — b x -b • Heine, Kutjelfunationen, t. i, p. 404. y Google 55.] EQUATION OF FUOHSIAN TYPE Writing the equation becomes which is the equation* of the parabolic cylinder. Case (vi). The equation is w"+ w'+7 ^7W = 0: when we take the equation becomes da? This corresponds to no particular equation in mathematical physics : it will be recognised as a very special instance of equa- tions most simply integrated by definite integralsf. Ex. Discuss, in a aimilar manner, the limiting forms which are obtained when the singularities of (i) the equation determined by Riemann'a P-function, (ii) Lamp's equation, expressed as an equation of Fuchsian type, are made to coalesce in the various ways that are possible. Equations with Integrals that are Polynomials, 56. There is one simple class of integrals which obey the condition of being everywhere regular, so that their differential equations are of the Fuchsian type ; it is the class constituted by functions which are algebraic. We shall, however, reserve the discussion of linear differential equations having algebraic inte- grals until the next chapter ; and we proceed to a brief dis of a more limited question. y Google 166 EQUATIONS WITH [56. We have seen that an equation of the second order and of Fuchsian type can be transformed to By = 1^" + (?„--i/ + G„^y = 0. Its integrals are regular in the vicinity of each of n singularities and of infinity ; the question arises whether the coefficients in the polynomials ©„_, and (?„_3 can be chosen so that one integral of the equation at least shall be, not merely fi"ee from logarithms or even algebraic, but actually a polynomial in s. This question has been answered by Heine* ; the result is that &,^i can be taken arbitrarily, and Q.^^ has then a limited number of determ- inations. If the above equation, in which G„_i = CoS"-' + CiS"-^ + . . . + C„_5Z + C^i , (?„_3 = k^-^ + h^"^ + . . . + k^^ + ^„_2, is satisfied by a polynomial of order ni, say by y ^ g^"^ + gi^^'"- + ■■• +9^-1^ + 9,^, then 80 that there are m + n — l relations among constants. The form of these relations shews that gi, g^, ..., gm ^re multiples of ff,,'- to express these multiples, m of the relations are required, and when the values obtained are substituted in the remainder, we have n — 1 relations left, involving the constants c and k. Assuming the points ai, a^, ..., a„ arbitrarily taken, and the coefficients Co, Ci, .,., c„_] arbitrarily assigned, we shall have these n—1 rela- tions independent of one another, and therefore sufficient for the determination of the K — 1 constants /c^, ^i, ...,^»_a. The first of these relations is m(m-l) + o,m + k, = 0, so that ks is uniquely determinate. Denoting by [kuh. ...,K]r the generic expression of a function of ki, k^, ..., k^, which is polynomial in those quantities, and the terms of highest weight in * Heine, Kugel/unctioneii, t. i, p. 473. y Google 56.] POLYNOMIAL I^fTEGRALS 167 which are of weight r, when weights 1, 2, . . . , u — 2 are assigned to ^1,^2, ..., fca-s, we have, from the m relations next after the first, for )■= 1, 2, .... m. When these are substituted in the remaining w — 2 relations, we have for s=l, 2, ..., « — 2. These determine the m — 2 constants fci, ^2, ...,kn-2', the number of determinations may be obtained as follows. Writing the equations become n - 2 equations to determine w — 2 quantities Xi, (c^, ,.., «B-a. In these quantities, the equations are of degrees m + l,m + 2, ...,m+m-2, respectively; and therefore the number of sets of values for iCi, x^, ..., Xn-2 is (m+l)(m + 2)...(m + «-2). But the same value of k^ is given by two values of x^, inde- pendently of the other constants k; so that the sets of values of «i. a's, ■•-, iTn-amust range themselves in twos on this account. Similarly, the same value of k^ is given by three values of iCj, independently of the other constants k ; hence the arranged sets of values must further range themselves in threes, on account of kj. And so on, up to k^^. Hence, finally, the number of sets of values of ^1, ..., kn--^ is (m + l)(m + 2)...(m + «-2) 2.3...JI-2 ^ (ffi + H-2)l ~ m 1 (n - 2) I ' which therefore is the number of different quantities Gn~i per- mitting the equation to possess* a polynomial integral of degree in. * In Gonnection with these equationa, a memoir by Humbert, Jourw. de I'Ecole Polytechaique, t. jliix (1880), pp. 207—220, may be consulted. y Google 168 POLYNOMIAL INTEGRALS [56. This result is of importance, as being related to those special forms of Lame's differential equation which possess an integral expressible as a polynomial in an appropriate variable. This polynomial can be taken as one of the regular integrals belonging to each of the singularities ; the other regular integral belonging to any singularity is, in general, a transcendental function and, in general, it involves a logarithm in its expression. Es!. I. Shew that a, linear equation of the third order, having all ita integrals regular, caii, by appropriate transformation of its dependent variable, be changed to the form where >/.=(^-a,)(e-<i,)...(E-<i.), Ill, itj, ..., On being all the singularities in the finite part of the i-plane, and ■where P, Q, B are polynomial functions in z of degrees ii-l, 2ii-2, 2ji— 3 leepectively. Shew that, if P and § be arbitrarily chosen, R can be determined so that one integral of the equation is a polynomial in i ; and prove that the number of distinct values of B is (m + 2»-3) ! m\(,2n-sy.' where m is the degree of the polynomial integral. Es. 2. Determine the conditions to be satisfied if has two distinct polynomials as integrals, so that every integral is a poly- nomial. Ex. 3. Determine how far the constants in the equation may be assumed arbitrarily if the equation is to possess two polynomial integrals. Sc. i. Prove that the equation /wg+l/'w|-t(.(.+ i)«*l»-o ■where »i is an integer, f(x)~x^+a3fi+ba:+c, and a, b, c are constants, admits of two integrals whose product is a polynomial in x. Ex. 5, Shew that the only cases, in which the differential equation of the .(i--)2+(v-(.+f.+i)«)*-.»=o y Google EXAMPLES 169 s whose product is a polynomial in ie of degree n, are as follows. If n is an evea integer, then either a=-^n; or j3=~Jn; or a+/3=-m, and y = ^, or -^, or -|, ..., or -re+i. If ra is aJi odd integer, then either a=-\n and y=i, or -|, or -f, ..,, or -|w+I, or ft or 3~1, ..., or »^\{n~\); or fi=-\r<, and 7=^, or -i, or -f, ..., or -i»i+l, ora,ora-l, ...,or<.-i(™-l); or a4-j3= -«, and y=i, or -i,or -§,..., or -)i+^: (Markoff.) Sr. 6. Sliew tliat, if the square root of a polynomial of degree m can be an integral of tte equation (V- 2 Xrf^,)^'""^+''i^" ^+. .+««-2 n^(;^-0 whore the exponents X and p. are subject to the usual relation, one of the exponents \„ ^„ say X„ must be half of a non-iiegative integer, this holding for each value of s ; also ^ra — 2X, must be a non-negative integer; and one exponent of the singularity at iofinity must be equal to - ^»!. If these conditions are satisfied, how many such equations exist? n Vietk.) Ex, 7. If the differential equation "■" -"•''"• 11 (»-.,) where ^(^) is a polynomial, the constants a are real and positive, and tlie Constanta e are real and distinct from one another, be satisfied by a poly- nomial ^ {x), then all the roots of (.c) are real, and no toot is leas than the least or greater than the greatest of th.e quantities e. (Stieltjes ; BSoher.) Equations with Rational Integrals, ■37. The investigation in § 56 suggests another question: what are those linear equations, all the integrals of which are rational naeromorphic functions of 3? Let a,, ..., Oto he the singularities in the finite part of the plane; let a^t, «^, ■•-, a«r be the roots of the indicial equation for Of-, and let /3i, ..., 8n he the roots of the indicial equation for z — x>. If every integral is to be a rational function of z, all the roots a,r, Owi ■-■, a„r must he integers; as no integral is to involve a logarithm, no two of them may be equal. Let the arrangement y Google 170 EQUATIONS WITH [57. of these roots be in decreasing order of the integers. The integral belonging to the index a,r involves no logarithms ; in order that the integrals belonging to the indices a„, a^r, ■■■, a»r respectively may involve no logarithms, H-2+...+(n-l), that is, \n{'ii—l), conditions in all must be satisfied, these conditions being as set out in § 41. Corresponding conditions hold for each of the singularities, and also for «= « ; so that there i»(«-l)(m + l) conditions of relation among the constants of the equation, in addition to the necessity that the indicial equation of each singu- larity shall have unequal integers for its roots. These conditions are certainly necessary ; they are also suffi- cient to secure that any integral of the equation is a rational function of z. For considering the vicinity of a,, each integral in that vicinity is of the form where «„,. is the least of the roots of the indicial equation, and Pm(3 — t,) is holomorphic in the vicinity of a,., for m= 1, ..., «; when m = «, P{z —Of) does not vanish, and for all other values of m it does vanish. If then o„r be zero or positive, the point s = », is an ordinary point for every integral in the vicinity of «, ; if Ojir be negative, then a,, is a pole of some integral, and it may be a pole of several or of all. As this holds in the vicinity of each of the singulaiities and of s = 00 , it follows that, in the vicinity of every singularity of the equation, including z — ca, every integral is uniform and has that singularity either for an ordinary point or a pole ; moreover, every integral is synectic in the vicinity of every other point : hence* the integral is a rational function, which is a polynomial if oo be the only pole. Thus the conditions are necessary and sufficient. It has been seen that the indicial equation for each singularity of the differential equation must have unequal integers for its roots. When these are assigned arbiti-arily, subject to the one relation (Ex, 2, § 46) which they are bound to satisfy, they amount y Google 57.] RATIONAL INTEQEtAXS 171 to (m + 1) K - 1 conditions ; so that the total number of necessary conditions is ^n{n-l){m + l) + (m + l)n-l = in(n + l)(m + l)-l. If such equations are being constructed, they are necessarily of the form where -^ = {z -a^ ...{z~ a»), and 0^ is a polynomial of order not greater than r{m— 1), for r= 1, ...,«. Henue the total number of disposable constants is m, from the positions of the singularities, + S \r{m-\)+ 1}, from the constant coefficients in Gi, ..., G„, that is, i»(n + l)(m-l) + » + ». constants in all ; and therefore, in order that the equations may exist, we must have \n(n + \){m-\) + n + m>\nin + X)(m + l)-l, so that m^K^-1. In obtaining this result, an arbitrary assignment of unequal integers as roots of the indicial equations has been made : and it has been assumed that these conditions are independent of the necessary conditions attaching to the coefficients, in order that the integrals of the equation may be free from iogarithms. It may, however, happen that a particular assignment does not leave all these conditions independent of one another, so that we might have i»(» + l)(».-l)+« + m.in(»+l)(m+l)-l-\, and therefore m = 7i'-l-\, and still have the equation determinate. An instance is furnished by the equation y Google 172 EQUATIONS WITH [57. which, although it has only one singularity in the finite part of the plane, so that m = 1, i! = 2, has an integral Ax"^ + Bx. For the most general case, however, we have JSr. 1. Investigate all the oasea in which the differential equation of the hypergQometric series haa every integral a I'ational function of the independ- ent variable. fiiir. 2. When the equation is of the second order, and all the assignments of integer roots are quite general, the smallest value of m is 3. Let the singularities be «[, ..., %,, with exponents aj, S,; a^, |9ji ...; a^, ft„; and let the exponent* for ^= a> be a, S- Choosing in each case the smaller of the two indices Or and (9r, let it he o,., for r = l, ..,,nt; then writing \r = »,-ar, a+ S n^ = <r, 3+ S o^=r, we have (§ 52) cr + r+ S \^m-\, which is the necessary relation among the exponents. Writing so that y also is a rational ftioction of z, our equation m y becomes say and here the integers Xj, X^, ,,., X,„ are, each of tliem, equal to or greater than Substituting, in the vicinity of a^, the expression (i-a.)^Dy=c,0{e^K)A provided c„(C+«)(d + m-X,.)-Hc„ g(«,) and the summation for a is for s=l, ..., m except s—r. As X^ is a positive integer, and thus is the greater root of tlie modified indicial equation, there is y Google 57.] RATIONAL INTEGRALS 173 one regular integral belonging to the exponent X^, whicii is a constant multiple of = y, say, where y^ = c^-^Cff, when 5 = A,, When we write and solve the equations for v^, o.^, ..., we find K{&) "' /(e+i).../(d+ur"- We know (§ 41) that there is a single condition to be satisfied in order that the integral belonging to the exponent may be free from logarithms ; as f(,6+n) vanishes to the first order for 6=0 when w=Xr, the condition is There i^ i jrrespundmg cond tion fjr eaih of the sm^ulirities ajii foi 4, = =o , s) thit we haie in + 1 condit una, wtii-h miohe the aibitrary constants ;! ^n 3 ind t.he positions of the singularities ds well as the assigned integers \ ^^ a- t Keeping the latter arlitrary, we aee that there m Lst 1 e at least three singularities in the finite part ot the plane when there are onlj three we oltiin a limited number of determinations of the equation it there aie ''+jo then p elements are left aibitrary among an otherwise limited number of determinatiins Df the equation* As the oquition is of the setond eider it is possible to plotted otherwise Assuming that the integial J" which bekngs to the exponent A^ f the singularity a^ la known, and denoting by ^the mtegiil whuh 1 elongs tc the exponent cf the same singulaiity welia\e rz"- Y"z+{yz'-rz) 2 i^=o, so that and therefore d (Z\ , 1 ™ , ,i -1 When the right-hand side is expanded in powers of s—a^, the first term involves {e—ar)~^~^', that is, the indes is negative. If ^ is to be free from Ic^arithms, the term in in this expansion must have its coefficient equal to zero — a condition which must be the equivalent of • The hypergeometric case indicated in the preceding example is given by »,.x.....=x„-i, a. ,,[,-„,). ..if-„j. which will be found to satisfy the conditions for a,, ... , a„ given in the text. y Google CHAPTER V. Linear Equations of the Second and the Third Orders POSSESSING Algebraic Integrals. 58. The general form of equation, having all its integrals regular in the vicinity of each of the singularities (including oo ), has been obtained ; in the vicinity of a singularity a, each such integral is of the form (,^-arH>,+ 'hiog{.~a) + i,,\kgU-a)]' + ...+4,.\hg{z-a:)]-l where each of the functions ^n, 0,, ..., 0, is holomorphic at and near a. In general, each of the functions is a transcendental function in the domain of a: they are polynomials only when special relations among the coefficients are satisfied. When attention is paid to the aggregate of the integrals so obtained, it is to be noted that the branches of a function defined by means of an algebraic equation belong to this class. If algebraic functions are to be integrals of the differential equation, they constitute a special class ; special relations among coefficients of the differential equation must then be satisfied, and, it may be, special restrictions must be imposed upon its form. Accordingly, we proceed to consider those linear equations whose integrals are algebraic functions, that is, functions of s defined by an algebraic equation between w; and z. It has already been proved (§ 17) that each root of such an algebraic equation of any degree in iv satisfies a homogeneous linear differential equation, the coefficients of which are rational functions of z. If the algebraic equation were resoluble into a number of other algebraic equations, neces- sarily of lower degree, each such component equation would lead to its own differential equation of correspondingly lower order; accordingly, we shall assume that the algebraic equation is irre- y Google 58.] LINEAR SUBSTITUTIONS 175 soluble and proceed to consider linear ditFerential equations whose integrals are tiie roots of an algebraic equation, , In the most genera! case, the degree of the algebraic equation is equal to the order of the diffei'entiai equation ; in particular cases (| 17, Note 1) it can be greater than the order : and as we seek algebraic inte- grals, it may be expected that these particular cases will occur. The investigation can be connected with an equivalent problem that arises in a different range of ideas. It has been proved that, given a fundamental system Wi, w^, ..., w^ of integrals of a linear equation of order m,, the effect upon the system, caused by the desciiption of a closed path enclosing one or more of the singu- larities, is to replace the system by another of the form Wm = OmlWi + Oj^W.^ + ... + 0,„mWm I say (iU]' Wm') — S{Wi, ...,w,„), where S denotes a linear substitution. By making the inde- pendent variable describe an unlimited number of contours any number of times, we may obtain an unlimited number of linear substitutions ; and so each integral could, in that case, be made to have an unlimited number of values. If, however, the fundamental system is equivalent to the m roots of an algebraic equation, then each of the integrals can acquire only a limited number of values at a point which are distinct from one another: that is, there can be only a limited number of substitutions in the aggregate. When therefore we know all the groups of linear substitutions in m variables which are of finite order, only those linear differential equations which possess such groups need be considered. Accordingly, if we proceed by this method, it is necessary to construct the finite groups of linear substitutions. Further, it is clear that the investigation can be associated with the theory of invariantive forms ; for the relations between w/, ..., Wju' and w,, ..., w^ constitute a linear transformation of the type under which these invariantive forms persist. Indeed, it was by this association with binary, ternary, and quaternary forms that the earliest results, relating to linear equations of the orders two, three, and four, were obtained. Some brief indications of this method will be given later (^ 69 — 72). y Google EQUATIONS OF THE SECOND ORDER [59, Klein's Method tor Equations of the Second Order. 59. The determination of linear equations of the second order, whose integrals are everywhere algebraic, is effected by Klein*, by a special method that associates it with the finite groups of linear substitutions of two homogeneous variables. Let w, and w, denote a fundamental system of integrals for the differential equation ; and let Ifi = awi + jSws, JTs = 7W, + Swj , be any one of the linear substitutions, representing the change made upon the fundamental system by the description of a closed path. Then taking Wj' the quotient of two algebraic integrals, so that s itself is an algebraic function, we have W.^ys + B' thus s is subject to a homographic substitution. Accordingly, the determination of the finite groups of linear substitutions in the present case is effectively the determination of the finite groups of homographic substitutions. Let any such group containing N substitutions be represented by t.W. t.(«) +«(«). and let t^,, (s) = s, the identical substitution : every possible com- bination of these substitutions can be expressed as some one of the members of the group. Take a couple of arbitrary constants a and b, subject solely to the negative restrictions that a is not equal to ■<frr(b) and b is not equal to -^((a), for any of the values 0, 1, .,., N ~1 of r and of s; and form the equation ^a(s) — a ifrj (s) - a t^tj^., (s ) - a _ „ Ms)~b■^ir,{s)-b ir^-d'^)-b * Math. Ann., t. ii (1877), pp. 115—110, ih., t. xu (1877), pp. 107—179; Varlemngen Hberdai Ikosaeder, {LeipKig, Teubaer, 1S84), pp. 113—123. y Google 59.] wrrn algebraic integrals 177 which is an algebraic equation of degree W^ in s. It is unaltered when s is submitted to any of the substitutions of the group ; for such a substitution only effects a permutation of the various N fractions on the left-hand side among one another. Hence, if any root s be known, all the N roots caa be derived from it by submitting it to the JV substitutions of the group in turn. For quite general values of X, the iV roots of the equation are distinct; but it can happen that, for particular values of X, a repeated root arises, of multiplicity v. From the nature of the equation in relation to the group of substitutions, it follows that each distinct root is of multiplicity c, so that there are N—v distinct roots. To consider the effect of this property of the equation, let the latter be changed so that the numerator and denominator are mulfcipKed by the denominators of i|'i(s), ..., ^fr^^l(s). It thus Can be expressed in the form where G (s, a) is a polynomial in s of degree N", the coefficients being functions of a, and G {s, J) is a similar polynomial, its coefficients being the same functions of b. Let X, be a value of X, such that 5 = <t, is a root of multiplicity v^ when X = X^ ; then the equation G(s,a) 0{a,,a) _ Q\s, h) G(cr„ b) ' N has — roots each of multiplicity v^ when X =X,. But each such root is a root of multiplicity iij ~ 1 of the equation d [ gfoo) gfa, ■») )_■ '<u\e{s.bj G(r„6)( ■ that is, of the equation AW = fl(»..)^-«it^-<J(..)«(''-A) = 0; y Google 178 Klein's method for [59. of fclie roots of this derived equation. Moreover, we then have G(s,b}G{<y„b) = X-X,. Let X^ be another vahie of X, suoh that s = cr^ is a root of the equation of multiplicity j-j when X = X^. A precisely similar argument shews that each distinct root of the equation is of multiplicity v^; that there are N-^v^ distinct roots; that each such root is of multiplicity v^—l for the equation A (s) = ; that these roots account for f(.-i) of the roots of the derived equation ; and that we have O/' -r x N where 'I's is a polynomial in s of deg;ree ■— . Proceeding in this way with the various values of X that lead to multiple roots of the initial equation, we shall exhaust all the roots of the equation A (s) = 0. The degree of A (s) is 2jV — 2 ; for if G(s.a) = s^Ma) + s''-'Ma)+ ..., then B{s.b)-^/.(b) + s''-'Mb)+...; and therefore 4(.)-»'"l/.(«)/>(6)-/.(4)/.(«)) + -. But taking account of the roots of A (s) = 0, as associated with the multiple roots of the original equation for the respective values of X, we see that its d and therefore y Google 59.] EQUATIONS OF THE SECOND OKDEIi 179 Each of the integers v is equal to or greater than 2, so that each of the quantities 1 is equal to or greater than J. Hence the smallest number of different integers i- is two ; if there were only one, the left-hand side would be < 1, while the right-hand side is > 1. The largest number of different integers p is three ; if there were four or more, the left-hand side would be equal to or greater than 2, while the right-hand side is less than 2. In the first place, let there be only two integers, y, and v^ ; then 1 J. __2 From the nature of the case, v, < iV, v^ ^ JV, so that hence the only possible solution is :,. = JV, v, = N; (I), and N is an undetermined integer. In the next place, let there be three integers, v^, v^, v, : then 111,2 Vi Vi 1-3 JS At least one of the integers v must be 2 : for if each of these integers were ^ 3, the left-hand side would be < 1, while the right-hand side is >1, as iV is a finite integer. Taking j',= 2, we have 11,2 Another of the integers v may be 2. Let it be v^\ then N = ^V3, and we have the solution v, = 2, v,^2, v, = n, JV=27i, (II), where n is an undetermined integer. If neither of the integers p^ and v^ be 2, one of them must be 3 ; for if each of them were ^ 4, then - + —^^, and so y Google 180 ALGEBRAIC INTEGBALS AND [59. could certainly not be equal to ^ + -^ . Taking v^ — 3, we have jV N' so that Cj < 6 : thus possible values of v^ ai-e 3, 4, 5. The solu- tions are j,, = 2, r, = 3, v^=% N=12, (Ill), „j=2, ^3 = 3, i; = i, N=2i, (IV), v, = 2, v^^S, v,= 5, iV=60 (V). 60. The finite groups are thus known ; the corresponding equations in s are required. The solutions will be taken in order. I. Instead of X, we take a quantity Z, defined by the rela- tion , x-x, so that Z=0 gives X — X^, that is, gives s = Si, a root repeated N times, and Z—xi gives X = X^, that is, gives s = s^, a root repeated N times. We have J V ('-'■)•• ' e(s, l>)G(j„i)' X-X,~^i^,: and therefore absorbing the constant (? (Sj, b)-r-G (s^, b) into the variable Z. 11, III, IV, V. These cases are of the same general form. Instead of X, we take a quantity Z, defined by the relation X-X, X,-X,' then 7=0 gives j: = X„ 2.1 gives X = X,, 2=« givesX-X,, and thus Z:Z-1 : 1 = (X - X,) (X, - X.) : (X - X,) (X, - X.) : (X - X.) (X, - X,). y Google 60.] POLYHEDRAL FUNCTIONS 181 But ' G(s,b)G(aub)' Y Y - '^°^' ^ G{s,b)G(^,,b)' ^~-^'^G{s,b)G(^,,by and therefore ^ : 2-1 : 1 = ^*/^(s) : B<t',-'(s) : ^."'(s). where A and B are constants which, if we please, may be absorbed into the functions 3>s and <I», respectively. Now these groups are the groups that occur in connection with the polyhedral functions* : and the polyhedral functions can be associated with the conformal repreaentation-|-, upon a half-plane, of a triangle, bounded by three circular arcs and having angles equal to - , - , - , The analytical results connected with these investigations can be at once applied to the present problem. Denoting derivatives of Z with regard to s by Z', Z" , Z"', ..., we have (T. F., § 275) Z'\Z' ^'\Z')\ ^ or, taking account of the properties^ of the Schwarzian derivative, we have ' ■ ' Z- '^ (2-1)" Z{Z-l) The forms of the functions for the various cases II, III, IV, V for II, * T. P„ g§ 276—279, 300—302. t T. F., %% 27*. 275. + See Ex. 3, § 62, of my Treatise on Differential Equations. y Google 182 ALGEBRAIC INTEGRALS OF [60. for III, Z:Z-\ :1 = (s* + 2sV3 - 1)= : lav's sHs* + 1)^ : (s^ - 2s^V3 - 1)= ; for IV, Z:Z~l :1 = (s^ + 14s* + l)»:(s^=--33s'-33s'+l)^: 1085^(5'- 1)= ; and for V, Z.Z-1 : 1 = (s^~~ 228s" + 4-9is"' + 228s' + If : {s*=+ 1 + 522s''(s™ - 1) - 10005s'Hs"' + l)p ;-1728s«(s" + lls°-l)=. These results* can be obtained by purely algebraic processes, from the properties of finite groups proved by Gordanf. 61. These results can be applied at once to the determination of linear equations of the second order <Pw dw all the integrals of which are algebraic. Denoting the quotient of two integrals Wi and w^ by s, we have§ w =5'"*se"*^'^ w =s''^e'^'^'^, w.,s = w say. As all integrals are to be algebraic, it follows that s and s'-S are algebraic ; accordingly, fpdz must be the logarithm of an algebraic fv/nction, which is a Jirst condition. Further, in the equations under consideration, both p and q (and therefore also 2/) are rational functions of z ; and therefore [s, z] = rational function of z, * They are sliglitly changed front the forms in % 302, g 278 {I.e.) ; the ciiange is made, eo aa to associate the indices v.^, t^, y.j with the values Z = 0, Z — 1. Z = rrj respeo lively. t Math. Ann., t. m (1877), pp. 23^6. See also Cajley's memoir, "On the achwaraian derivative and the polyhedral functions," Coll. Math. Papers, t. Si, pp. 148—216. g See my Treatise on Differential Equaiions, §g SI, G2. y Google 61.] EQUATIONS OF THE SECOND ORDER 183 and the quantity s is subject to the transformation of the finite group. Now we have seen that \'.^ F~+ ik-if + Wz^ ' in cases II, III, IV, V ; and for case I, it is easy to verify directly that From the properties of the Schwarzian derivative, we have hence, taking account of the particular form of [s, Z] which is actually known, and of the generic form of [s, z] which ia required, we see that, in order to satisfy the conditions, we must have Z=R{z), where K is a rational function of z. Conversely, the conditions will I if .Z' is any rational function of z. Accordingly, the <,l equation oftJie second order viust have the coefficient of It/ in the form where u is om algebraic function of z ; and its invariant I{z), which is q — \P' — h'j^' '"''"^* ^s of the form 1--. — + ^*^- ^{Z-\Y^ Z{Z^i) \\Z,z}, where Z is any rational function of z ; the integers v-^, v^, va in the first form are the integers of the finite groups in cases II, III, IV, y Google 184 MODE OF OBTAINING [61. V ; and N in the second form is an integer. When these con- ditions are Batisfied, the integrals are given by where, for the first form, s is determined in terms of Z, the rational function of z, by the equations at the end of § 60 ; and for the second form. Construction of an Integral, when it is Algebeaic. 62. The preceding investigation is adequate for the general construction of linear equations of the second order which are integrable algebraically ; there still remains the question of determining whether any particular given equation satisfies the test. When the equation is of the form d'vj dw inspection of the form of p at once determines whether it satisfies the condition which governs it specially. Assuming this con- dition to be satisfied, we construct the invariant I{z) of the equation, where and then, if the original equation is algebraically integrable, we must also have (2-1)' Z{Z-1) /W-l^^TT, +11^.^ ©■-il^^!. where ^ is a rational function of 2, and the integers c,, v^, vs belong to one of four definite systems. It may happen that the identification is easy, because Z has some simple value; the simplest of all is, of course, given by y Google 62.] ALGEBRAIC INTEGEAL9 185 Z= z. When the identification is not thus obvious, it is desirable to have a method of constnicting the rational function Z if it exists ; when it has been constructed, the further id entiii cation is only a matter of comparing coefficients. Should this identification be completely effected, then the integration of the equation is given by the results of § 60, Such a method is given by Klein*, who uses for the purpose a comparison of those terms on the two sides, which are connected with the poles and have the highest negative index. A rational function is detenninate save as to a constant factor, when its zeros, its poles in the iinite part of the plane, and their respective multiplicities, all are known ; and this constant factor is determ- inate, when the value of the rational function is known for any other value of the variable. Accordingly, let a denote a zero of ^ of multiplicity a, and so for all the zeros ; let c denote a pole of Z {and therefore also of Z— 1) of multiplicity y, and so for all the poles ; and let h denote a zero ot Z — 1 of multiplicity ^, and so for all its zeros : then U{z-ay U{b-c)y n{b~ay ■ n(z-c)y' where the multiplicity ^ of 6 is not used directly in the ex- pression. Consider now the right-hand side of the expression for I (i). In the vicinity of a, we have where t7 is a regular function oi 2~a, not vanisiiing wlien s = a; so tliat IdZ 'J „, . ^JJ-^3i + -'*<'-<■>. and the unexpressed terms in [Z, z] having exponents greater than — 2. In the vicinity of c, we have Z=(z-cyiV, Z-l=(z-c)-iV,. " Math. Aim., t. sii (1877), pp. 173—176; the espoaitiou given in the test does not follow his eiactlj, as he transforms the equation 90 aa to secure tliat s = o: ia an ordinary point. y Google 186 CONSTRUCTION OF AK [62. where V and V^ are regular functions of 2 — c, not vanishing when z=c; thus IdZ -7 „, 1_ dZ -, Z-ldz ^ z- _fS,{^-c), 1^. ^ -e)" ■■■' the unexpressed terms in \Z, s] having exponents greater than — 2. In the vicinity of 6, we have 2-l-(»-6)»If, where TT is a regular function oi s — h, not vanishing when z=h\ Ho that Z-\d-^ z-b* ^ °'' the unexpressed terms in [Z, z\ having exponents greater than — 2. We thus have taken account of all the highest terras with negative indices which arise through zeros or poles of Z and Z—1. On account of the form of [Z, z], which is Z' ^\Z') ' it is necessary to take account of the poles and the zeros of Z'. As Z is rational, all its poles are poles of Z' and the latter has no others; so that, on this score, no new terms arise, A repeated zero of ^ is a zero of Z', and all these have heen taken into account; likewise for a repeated zero of Z—1. Hence we need only consider those roots of Z', which are not repeated roots of Z or of if — 1 ; let such an one be t, of multiplicity t, so that z.(z-tYqu-e,. where Q is a regular function of ^ — (, not vanishing when s = t; then the unexpressed terms io [Z, s] having exponents greater than — 2. y Google 62.] ALGEBRAIC INTEGRAL 187 Gathering together the terms with the largest negative index, we have, for Cases II, III, IV, V, (s- ay (s - by {z - cf {z - ty where the unexpressed terms have integer exponents greater than — 2 ; and in this expression the significance of a, b, c, for the construction of Z, must be borne in mind. Actual comparison with the form of / {z) then gives indications as to which set of values of II,, Vi, Vj must he chosen, and determines the values of a, ^, 7- The construction of Z is then possible and, Z being known, the complete identification of the right-hand side with the known value of /(if) is merely a matter of numerical calculation. For Case I, we have and the method of proceeding is the same as before. In particular instances, it may happen that no terms of the type tr + JT- (s-tf occur : Z' then contains no roots other than the repeated roots of Z and Z — 1. An example is given by ^- 4^' Further, it may happen that a= v-^, or ^ = vi, or y^Vsi so that the corresponding value of z, viz. a, b, or c, is then not a singularity of the differential equation. And, in particular, if if = CO is not a singularity of the differential equation and there- fore also not a singularity of the integral, then, if the equation be integrable algebraically, the numerator of the rational function Z is a polynomial in z of the same degree as the denominator*. " Thin form of equation is discoased by Klein in the memoir already quoted (note, p. 186) ; reference should he made to it tor further developments. y Google 188 M:^. 1. The equatio] blo algebraically, For l% = l(l-^)> whence J.2 = 2; We thus have an instance of case II, when a = 2. All the conditions a satisfied : and thus (§ 60) the integrals of the equation are givon by fie. 3. Construct a linear differential equation of the second order in its normal form, such that the quotient s of two of its solutions is given by 108s»(si-l)a 43 ' Sx. 3. Consider the equation We have . ^ 2s>-8s=-153a-82 + 2 (a-l) ' the terms indicated constituting all the infinities of /(s) of the second order. First, it is clear that there is only one root of Z' other than repeated roots of Z and Z— 1 ; it is characterised by y Google 62.] Klein's method 189 If it wereposBibly an instance of case II with m = 3, then we must have ^^l_^^^=^,sothati., = 2, 3 = 1, h^i, i{^-^^-^> >',-3,y= I, <: = (>, i(l-5) = 3^. ., = 3,y = 2,«=-l; and therefore with the condition that 2—1 when z = h = i, so that A=i. But then shewing that Z' docs not possess a root z = l= 1 ; hence the example is not an instance of case II. If therefore the equation is algebraically integrahle, it must bo an instance of case III. We must have therefore i{ 1 — ^)=A^i whence ff=l, b=i, and then, either Q = l, a = Q, y = % c=-l; Taking the former, we have from the poles and zeros ai Z\ s& Z= 1, when s = i, we have A = 2, bo that SO that Z- I has the roots 2 = 8, s= - i; hut y Google shewing that Z' does not possess the f. values is not possible. Taking the latter, we have ; and thus the first assignment of ■sof^; asZ=l when s-i, e have A =^, and then so that Z—\ has 2=i,z— -ifor roots, and Z" has 2=1 for a root. The preliroinavy conditions are thus satisfied ; it is easy to verify that this value of Z gives the coinpleto value of /(:). Hence, after the results of § 60, the intf^ral of the differential equation is given by the equations ^si + 2aV3-l Y _ {a-VVf \} 2s ' algebraically integrable. ^_ +<S-i*.)s-A-o, and bt n th ntegrals where f=(i-ai)(^-a2)(^-''s),^dX,=i{a-a),X,=iO-^),),3=My-/}i discuss the possibilities of algebraic integrability for the values \^h ^2 = f. \ = \- In particular, shew that, if %= - 1, «3=0, then y Google EQUATIONS OF THE THiRD ORDER Equations of the Thied Order with Algebraic Integrals. 63, When we pass to the considGration of hnear equations of order higher than the second which are algebraically integrable, the discussion can be initiated in the same way as for equations of the second order ; but the detailed devdopment proves to be exceedingly laborious, and it has not been fully completed for each case. Only a sketch will here be given Dealing in particular with the linear equation of the third order, we take it in the form where p, q, r are rational functions of s, subject to the limitations imposed by the regularity of the integrals in the vicinity of eafih singularity (x included). If w^, w^, w, denote three i independent integrals, we have (§ 9) BO that, as Wj, Wa, Ws are algebraic functions of s, it follows that p, a rational function of z, must be of the form where m is an algebraic function of s. This is a first condition : it is the same as for the equation of the second order (§ 61): and it is easily obtained as a universal condition attaching to any linear equation which is algebraically integrable. Now substitute for v) by the relation and let y^, y^, y^ denote the three integrals corresponding to M>i, Wa, Misi owing to the character of p and the functional character of the integrals w, the integrals y are also algebraic functions of z. Thus the equation in y, being /" + 3Qy + ii-0, yGoosle 192 EQUATIONS OF THE THIRD ORDER [63. where Q = q-p'-p' I R ^ r - Spq -i- ^p' - p" ) ' is to be algebraically integrable. Denoting by s and t the quotients of two integrals by a third, we have The quantities s and t are algebraic functions of z for equations of the class under consideration. The effect upon a fundamental system, when the independent variable describes a circuit enclosing one or more of the singulari- ties, is represented by relations of the form F, = a y, + b j/a + c Fj = «' ^1 + b' y^ -+ c' 7a = (t"l/i + &>, + c"(/3 J If S and T denote the corresponding integral-quotients, then „ ^ a' + b's + c't „ ^ a" + b"s + o "t a + bs + ci ' a + bs + ct Now if the equation is integrable algebraically, there can exist only a limited number of different sets of values of the integrals ; so that the number of sets Y,, ¥^, Y^ is finite, and the number of simultaneous values of S and T is finite. If then we know all the homogeneous linear gioups m three variables, or (what is the same thing) all the lineo-lineai groups in two variables, which are finite, then each such finite group determines its set of values of Yi, Y,, Fa and the set of values of S and T, and so it determines a linear equation the integrals of which are algebraic: and con- versely, each such linear equation is characterised by a finite group. 64. In order to utilise the method for the present purpose on the lines adopted for the equation of the second order, it is necessary to deduce from the differential equation certain differen- tial invariants involving s and *, these invariants being expressed in terms of Q and E. This can be done in two ways. It is clear that, as s implicitly contains five arbitrary constants, it satisfies a differentia! equation of order five ; and that, as ( is of the same functional form as s, it satisfies the same differential equation. y Google 64.] WITH ALGEBRAIC INTEGRALS 193 On the other hand, as s and ( combined contain eight arbitrary constants implicitly, it may be expected that the two differential equations, which they satisfy and which wili involve both of them, will be each of the fourth order or will he equivalent to two of the fourth order. The single equation is, for some purposes, the more important in the formal theory of the hnear equation, which will be left undiscussed ; for the present purpose, the two equa- tions prove to be the more important. Accordingly, we substitute sy, for 1/2, and tz/, for y^, 1 the equation I integral of this equation, > whence, remembering that y^ have 3s'y," + 3fi'>/ + {SQs' + s'") y, = 0| 3(>," + 2t'%' + (SQt' + t'") !/, = 0) Differentiating each of these once, and substituting for y,'" from the linear equation which it satisfies, we have Qs'%" + (4s'" ~ 6Qs') y/ + [s"" + SQs" + 3 (Q' - R) s'] 2/1 = 0) Qt"y," + (W" - GQt') y: + [*"" + 3Qt" + 3 (Q' - ii) t'} y, = 0\' so that there are four equations, linear and homogeneous in the quantities y", y-[, y,. When the ratios of y" : yl : y^ are eliminated from the first pair and the first of the second pair, we have -3(ii-Q') and when the same ratios a pair and the second of the se i likewise eliminated from the first >nd pair, we have ("", «'" 6!" -3Q »'", 3." 3«' 1'", 3i" 31' -3(E-Q') ('", 3f", 3(' These, in fact, are the two equatio satisfied by s and (. F. IV. 0. f , , s"\ 35", 3s' *"', 3(", 3(' , each of the fourth order, 13 yGoosle 194 INVARIANTS FOE AN EQUATION [64 Suppose now that two solutions (other than the trivial solu- tions, s = constant, ( = constant) are known, say Solving the first pair of the foregoing equations for y^ : y,, we have 3 (aW - o-V") y,' + (^'"t - a'r'") y, = 0, and therefore neglecting an arhitrary constant arising as a factor on the right- hand side. Hence a fundamental system of integrals of the original equation is {</'t n~ tWt'-c ■i. or the original equation can be integrated if two particular solutions of the equations in s and t are known. 65, Moreover, from the source of the two equations which serve to determine s and i, it is to be expected that, when the above two (being any two) particular solutions s— a-, t = T, are known, the complete primitive of the two equations is '/ + b'lT + c't ll+b(T+ GT t = where the constants a, h, c, a', V, c', a", b", c" are arbitrary so fe,r as those two equations are concerned. This result can be stated in a different form. The two equations in question can be written As"" + iBs"' + 6Gs" - SQ (As" + 2Bs') - 3 (-K - Q') As' = 0, Alf'" + 4Bt"' + 6Gt" - SQ (At" + 2Bt') -S{R- Q') At' = 0, where A, B,G are the three determinants in I, t"'. 3t", W I Its — s"" t - I ii" t" - s"t'". yGoosle 65.] OF THE THIRD ORDER 195 SO that ^ = 9mi, £ = -3%, 6' = 3^,; then solving the preceding equations for Q and for R — Q' in turn, we find 3e=!^'i-^-ig)'-/(M,.) ) and ' ' I, -2T(-S-Q-) = 9°'-6 °-'°'t*°' - + sf-)'-f(».'.^) Ill W[^ \wi/ ; say. The latter equations may he regarded as the equivalent of the two equations, which have been solved ; and therefore we may expect that / / «' + f>'s + c't a"+b"3+c"t ^]^j-(^ f ^-j r faf + b's + c't a" + 6"s +c"t \ r , ^ , \a + bs + ct a + bs + ct J \ ■ ■ " the actual verification, which is comparatively simple, is left as an esercisa Clearly these are generalisations of the property of the Schwarzian derivative, represented by [cs + d j The two invariant functions / and J were first indicated* by Painlev^ ; they subsequently were simpUfied to a form, which is the equivalent of the above, by Boulanger-f-. The invariance of the functions / and J, as indicated, exists for lineo-linear transformation of s and t. There is also an invariance for any transformation of the independent variable z ; for we easily find the equations I(s, t, z)^I{s, t, Z)Z-'+2{Z, z], J(s, t, z) = J(s, t, Z) Z''~9I (s, t, Z) Z'Z" ~ 9 ^ (^. ^1- where Z is any function of z. Also ^I'(3,t,2 ' Comptes Efndus, I. oiv (1887), p, 1830. + See his Thftse, Contribution h I'elude des ^q'uatiimB differentielies Un$a it lumioginei intSgrabUs algSbriquement, [Paris, Gautliier-Villarfl, 1897). 13—2 yGoosle 196 INVARIANTS AND [65. and therefore j(s, t, z)+^r{s, t. .) = [J(s, (, Z) + ^r{s. t, Z)-\7i\ is an invariant for any change of the independent variable s. Dropping a numerical constant, this is the function which is the known Lagnerre invariant in the formal theory ; that is*, if the equation be transformed, by the relation --(sr to the form then As the transformation --»f=(^.-ii)(SJ' 'dZ\-^ 1-' yii) leaves the quotient of two integrals transformed only as by a lineo-linear substitution, it follows that the preceding function, say L (s, t, s) = J is, t, z) + f 7' (s, t. z), is unchanged by lineo-linear transformations effected on s, t; also, except as to a factor Z'', it is unchanged by transformation effected on the independent variable. Now 30 that we have " See a paper by the author, Phil. Tram. (1888), pp. 383, 390, Lagnerre's invariant was first aanounoed in two notoe, Comptee Hendus, t. Lsixviii (1879), pp. 116—119, 224—227. y Google 65.] FINITE GROUPS 197 which is the full expression of Laguerre's invariant in terms of the derivatives of s and (, 66. The next stage is to associate these invariants with the algebraic equations in two variables, which admit of one or other of the finite groups. These groups have been obtained by Jordan* and Valentinerf ; and references to other writers are given by Boulangeri- A method of using the results is outlined by Pain- lev^g as follows. Let (s, t), 1^ (s, () denote two irreducible invariant functions of a finite group of order JV; the functions are given by Klein|| for the group of order 168, and by Eoiilanger (I.e.) for the group of order 216. As these functions are invariable for each substitution of the group, and as s, t are algebraic fimctions of s, it follows that and -i^ are rational functions of s, say </,{.,() = * (4 f(s,t)=^^{^). Conversely, taking * and ^ to be arbitrary rational functions of z, these two equations give rise to N sets of simultaneous values of s and t as algebraic functions of z ; and if any one set of values be represented by <t, r, all the others are obtained on transforming a- and t by all the iV - 1 substitutions of the group other than the identical substitution. These two equations are used to obtain the first four derivatives of s and ( with regard to s ; and with these derivatives, the two invariants I(s,t,^). J{s,t,z) are constructed. The functions so formed involve derivatives of <!> and ^ ; and the coefficients of these quantities are rational in the derivatives of <^ (s, t) and i^ (s, t). As 7 and J are invariantive for the group, the coeiEcients specified are rational functions of s and t, which must be invariantive for the group and are therefore rationally expressible in terms of ^ and t/t, that is, in terms of <& • Crelk, t. Lxxxiv il878), pp. 89—215 ; AUi delta Jl. Accad. di Napoli, t. viii (1879), No. II. t Kj^b. Videmk. SeUk. Skr., 6 E., t. v (1889), pp. 61—235. X In the Ttese, already cited on p. 19S, note. % Comptei EendMS, t. civ (1887), pp. 1829—1832, ib, t. cv (1887], pp. 58—61. II Math. Ann., t. xv (1879), pp. 265-267. y Google 198 ALGEBRAIC [66. and ^, Thus I(s, t, e) and J{s, t, z) would be expressed as rational functions of s. Accordingly, taking 3Q = /(s,(,2), R = hI'{s,t.z)-i,J{s.t,z), we have the differential equation y'" + my' + Ry = 0. The earlier investigations shewed that its integi-als are expressible in terms of s, t, and their derivatives ; and we thus have a method of constructing all the linear differential equations of the third order which are integrable algebraically. There is a double arbitrary element for each group, viz. the arbitrary forms of the rational functions $ and 'P ; and there is a limited number of groups, 67. While this outline is simple enough in general descrip- tion, the application to particular cases requires extremely elabo- rate calculations. These have beeti effected by Boulanger for the group of order 216 ; they do not appear to have been yet effected for any one of the other groups. As, however, the enumeration of the finite groups in two quantities s aud ( is complete, the subject offers an. interesting, if a laborious, field of investigation. In the absence of the complete table of equations, for all the finite groups and for two arbitrarily assumed functions ^ and "^j it is not possible to use a method, analogous to that of § 62, to determine whether a given equation of the third order is algebrai- cally integrable or not; it is not even possible to recognise to which of the groups it would belong if it were algebraically integrable. Indications of two general methods of procedure have been given by Painlev4 and have been developed to some extent by Boulanger; but the methods, while general in description, suffer from the same kind of difficulty as the method indicated for the construction of the equations, for the calculations are exceedingly laborious. We have seen that, if two particular values of s and (, say t aud t, are known, then an integral of the differential equation is given by J/ = (o-'V---tV')-*. yGoosle 67.] INTEGRALS Hence, if we take so that the number of values, which u can acquire, is equal to M or to a Hubmultiple of N, where JV is the order of the associated group; let the number of values be n. Now if y ia algebraic, every zero of y and every infinity of y are of a finite order, which is commensurable in every instance ; and therefore all the infinities of u are simple poles with commensurable residues. Substituting for u in the ecjuation y'" + ^Qy +Ry = 0, we find u" + Suu' + w^ + SQu + R=0, a non-linear equation of the second order satisfied by u. This equation renders it possible to test the character of the poles and the residues of u. If these are of the appropriate type, then the equation is satisfied by a relation of the form where A^, Ai, ,.., An are polynomials in s, and A^ is the product of the factors corresponding to the poles of a. Then there is the further test that this algebraic function u must be such that is algebraic. Manifestly, the calculations will generally be too elaborate to make the method eifective in practice. Equations of tub Fourth Order. 68. As pointed out* by Painlev^ the processes just indicated can formally be applied to linear equations of any order: but of course, if any advance towards final conditions is to he made, it is necessary to know all the finite lineo-linear groups of transforma- tions in a number of variables less by one than the order of the ' Compls Eendua, i. cv (1887), p. 59. y Google 200 EQUATIONS OF THE FOURTH ORDER [68. equation. Towards this enumeration of groups in three varia- bles, which are associated with the linear equation of the fourth order, Jordan* has constructed a characteristic numerical equation which, when completely resolved, would indicate the order and the composition of each such group : but the resolution is exceed- ingly long and, owing to the number of cases that must be considered, it has not been completed. In these circumstances, no detailed results of a final critical character can be obtained for an equation of the fourth order or of any higher order : the only results obtainable are of a general character, and arise through the association of groups in general with linear equations. The equation of the fourth order, which may be written w"" + 4pMi"' + 6}W!" + 4rw' + swi = 0, can be transformed by ^g/p<i= ^ y into f" + eQy" + iUy' + 8^0. We denote a system of four integrals by j(i, (/,, y,, y^, and we introduce three quotients s, t, u, such that then s, (, u are simultaneous solutions of three equations of the fifth order in the derivatives. If a-, r, v are a special set of solutions, then yi = -i yi = yi<y, Vi = y^t, y^ = ^if ■ The complete primitive of the three equations is of the form s _ _ _t u a' + b'a- + c't + rf^ ~ tt" -i- 6'V 4- c"t + d'V ~ a'" + h"'<T + d"r + ' Atti della B. Accad. di Napoli, t, viii (1870). No. 11, p. 25; instead of dealing with lineo-luiear traasformations in three vacia.bles, Jordan deals ivith homogeneouB linear aubstitutioiiB in four variables. y Google 68.] WITH ALGEBBAIC INTEGRALS 201 There are three functions of the derivatives of s, t, u, with regard to z, which are invariantive for substitutions such as the precodiug relations expressing s, (, u, in terms of a, t, v; and they are equal to If the determinants S ± {8"'t"u'\ % ± (s"t"u'), S ± {s''t"'u'), S + {s't'V) be denoted by p, pi, p.^, p^ respectively, then say ; if, in addition, the determinants £ ± (s'Tu"), S ± (s-'f'u') be denoted by p^ and p^ respectively, then say ; and if the determinant 2 ± (s''s"'s") be denoted by p^, then S-so.= -& ,„„„,„„„ «-3;-8e-=f;-i87»*+2W. say. The three quantities I^ (s, t, u, z), /g (s, (, u, z), I^ (s, t, u, z) are unchanged when lineo-linear substitutions are effected on s, t, u; and the combinations /.+ 2/,--l/,"-A/,', are also unchanged, except as to a power of Z', when e is replaced by Z, any function of z. The proofs of these various statements are left as exercises. y Google 202 algebraic integrals and [69. Equations, having Algebraic Integrals, associated with Homogeneous Forms. 69. It has already {§ 58) been stated that the discussion of the equations, which have algebraic integrals, has been associated with the theory of homogeneous forms : the association can be seen to occur as follows. Using the preceding notation of §§ 63 — 66 for the quantities connected with any linear equation of the third order, we denote by s and ( the quotients of any two by the third out of any three linearly independent integrals of the equation If, then, all the integrals of this equation are algebraic, both s and t are algebraic functions of z ; they may therefore be regarded as determined, in the most general case, by a couple of distinct algebraic equations, say /.(»,(, »).0, /,(s,i, «) = 0, or by 9,{s.^) = 0, g,{t,z) = {). Eliminating z between the pair of equations in whichever form they are taken, we obtain a relation of the type i^i,(s, = 0, where Fg is a non- homogeneous polynomial in s and t, because it is the eliminant of two polynomials. Replacing s and i by jij -^ y-i and y^-i-yi respectively, and multiplying by the proper power of 3/1 to free the equation from fractions, we have ^ (yu y2, y^ = ^y, where f is a homogeneous polynomial in its arguments or, in other phrase, is a ternary form in 3/,, y,, y,. Further, the above form of equation is obtained from dht) , „ (iHv , „ dw , „ ■j T + 3jo , T- +3q-:j- + rw = 0, by the transformation y Google 69.] HOMOGENEOUS FORMS 203 and therefore that is, F{vii, Wj, M'i) = 0, on rejecting the i'actor e'"/?*', which occurs because F is a ternary form (say) of order in. Hence it follows that when tlie integrals of a liiiear equation of the third order are algebraic Junctions, a homogeneous relation of finite order exists among any three linearly independent integrals. Moreover, when any other set of fundamental integrals F,, Y,, Y, is taken, we know that y, = a,Y, + a,Y., + a^Y, y^=h,Y, + b,Y, + b,Y, where the coefficients a, b, c are constants. The variables in the homogeneous ternary form are therefore subject to linear trans- formation; and thus the theory of ternariants can be associated with those homogeneous linear equations of the third order, which have tbeii' integrals algebraic. The various cases will arise according to the order of the form F; this order is always greater than unity, because the integrals considered aie linearly independent. If, still further, we choose to combine the geometry of the ternary form with the form in its association with the equation, then the preceding algebraic relation ^ = is the equation of an algebraic plane curve referred to homogeneous coordinates r the curve is usually called the integral curve. equation of the fourth Wo order may proceed similarly with a n ec dw when all its integrals are algebraic. If we choose, we may trans- form it by the relation y Google 204 EQUATIONS OF THE the quantity e^^^'^ must be aigebraic, because where (7 is a non -vanishing constant ; and the equation in y, which is of the form dz' dz^ as ■' has all its integrals algebraic. Taking any four linearly inde- pendent solutions y-i, y^, 1/3, y,, and writing then as p, a-, t are algebraic functions of z, they must be given by three equations of the form or of simpler equivalent forms, which are completely algebraic in character. Eliminating z between the first and second, and also between the first and third, and taking the eliminants in a form free from irrational quantities if these occur, we have two equations F,{p,,,T)=0. ff.(p.^,T)=0, two non- homogeneous polynomials in p, a, r. Replacing these quantities by their values in terms of ^1,^2, 1/3, 2/4, and multiplying ea«h equation by the power of 1/,, appropriate to free it from fractions, we find where F and G are homogeneous polynomials in their arguments or, in other phrase, are quaternary forms in 7,, y^, y^, ya- As in the case of the cubic, these equations imply the fiirther equations F(Wj, Wa, Wj, W4) = 0I so that, when the integrals of a homogeneous linear equation oj the fourth order are algehraic Junctions, two homogeneous relations of finite order earist among any four linearly independent integrals. y Google 69.] FOURTH ORDElt 205 Again, when the variables i/j, y-s, y,, y^ are replaced by any other set of fundamental integrals F,, Y^, Y^, Fj, the two sets of variables are connected by homogeneous linear relations; and thus the theory of quaternariants can be associated with those homogeneous linear equations of the fourth order which have their integrals algebraic. The various cases will arise according to the orders of the forma F and G; these orders are always greater than unity, because the integrals y,, y^, y^, y^ are linearly independent. We may also combine the geometry of quaternary forms with the forms themselves as associated with the equation. In that case, each of the equations F= 0, G = is the equation of a non-planar surface in three dimensions referred to homogeneous coordinates : the two equations combined determine a skew curve, which ac- cordingly is the integral curve. Similarly, in the case of equations of the fifth order, of which all the integrals are algebraic, we have three homogeneous non- linear relations among any fundamental set of integrals ; and there are corresponding associations with the theory of homogeneous forms in five variables and the allied geometry. And so also for linear equations of higher orders. Note 1. There cannot be two homogeneous relations among a set of three linearly independent integrals of an equation of the third order: for they would determine a limited number of sets of constant values for the ratios y, : y^: y^, contrary to the postulate of linear independence. Similarly, there cannot be three homogeneous relations among a set of four linearly independent integrals of an equation of the fourth order; for their existence would imply a corresponding contradiction of the same postulate. And so for other equations of higher ordei-s. It might however happen that, for an equation of the fourth order, only a single homogeneous relation exists among four linearly independent integrals; that, for an equation of the fifth order, the number of homogeneous relations among a fundamental set of integrals is less than three ; and so on. If the relations thus given in each of the respective cases are the maximum number of homogeneous relations that can exist, we can infer that not all y Google 206 BINARY [69. the integrals of the respective equations are algebraic: and a question arises as to the significance of the respective relations. iVofe 2. The converse of the general argument must not be assumed valid : that is to say, the existence of a homogeneous rela- tion between the members of a fundamental system of integrals of an equation of the third order is not sufficient to ensure the property that all the integrals are algebraic. Thus we know that a number of transcendental functions of a variable can be connected by algebraic relations : and such instances are not the only possible exceptions. 70. The preceding method of a.ssociating the theory of forms with linear equations does not apply directly when the equation is of the second order : for a homogeneous relation between two integrals would imply one or other of a limited number of con- stant values for the ratio of the integrals, which accordingly could not be linearly independent. This deficiency, however, is rendered relatively unimportant, because Klein's method explained in §1 59^62 for the equation of the second order gives the complete solution of the question propounded as to the cases when ail its integrals are algebraic. The results there given can be (and have been) obtained by processes directly connected with the theory of binary forms. After the preceding exposition, the analysis is mainly of formal interest, and adds little to the knowledge of the solutions regarded as functions of the independent variable. It will be sufficiently illustrated* by one or two examples. Ex. I. We tako the differential equation in the foi'm and consider the value of a homogeneous polynomial function of two integrals ^1 aud y^i linearly independent of one another. Let this polynomial be of order re, and write * For fuller diBOUsaion and details, see Faaks, Crelle, t. Lixii (1376), pp. 97— 142, f6.,t. Lxxxv (1878), pp. 1—25; Briosclii, itfalft, ^nn., t. zi (1877), pp. 401— 411; Forsyth, Quart. Journ., t. ixm (1889), pp. 45—78. A memoir by Pepin, "Methode pour obtcnirlesintfigralesalgSbriques des Equations dififeentialles lin^aires du seeond ordre," Row. Ace. P. d. N. L., I. isxiv (1883), pp. 243 — 389, may slso be consnlted with advantage. y Google 70.] FORMS 207 saj, where s is the quotient ^,-^-^3. When substitutioa is made for i/j and ^2 ill terms of x, let the vahie of/ be <p (x), so that Now \i R{yi,y^ = H{f) be the Hessian of/, and if H{u) ho the Hessian of a, so that ^W"-<-'>-(»S-<»-')(S)'}- We have also *! *■ ■"&;- y,SJ-eon.i say, so that -■£=^ Now j,-«-*Wi • *, , c 1 <;» ^S <^ Differentiating, and aubatituting for the second derivative oiy^, we have y^\dx J j/ dx u ds y^ d^ ds^ Multiply by n, and add the squaj^s of the aides of the preceding equation : ^n^21 L d^logu) \_ fdu\^\ _ I /d<p\\ d^{los<i>) The coefficient of CVa"' on the left-hand side is 60 that the Heaaian in terms of functions of x : let this be written y Google 208 EXAMPLES [70. If now * (y^, y^) denote the oubiuo variant of/, so that 1 rofdH of a//) then, proceeding in a similar way, we find And so for other covariants. As a special case*, let it be required to find tho value of 0, if when the binary form is tho quadratic a„ylH2a^3/lya + a32'2^ c^ {x) is a root of some rational function of x. In this instance, a constant ; hetice i^ {:c) is either a rational function, or is the square root of a rational function. The integration is immediate ; for d$ Cdx a^s^ + ^a^s + a^ if>ixy The value of s is thus known : and the consequent values of y^ and y^ a immediately given +. .Ec. 2. Shew that, if the integrals of the equation and is a root of some rational function of x, then <^* must be rational ; and obtain tho relation between / and if) (x). Ex, 3. The integrals of the equation and (a;) is a root of some rational function of x ; shew that, unless ^ {x) is actually rational, the quadrinvariant of the binary quartic must vanish. In either case, find the relation between I and 1^ {x). (Brioschi.) * Fuchs, CreiU, t. lxxsi (1876), p. 116. t See my Treatise on Differential Equations, % 62. y Google 70.] OF BINARY FORMS Ex. 4. Find the value of / in the equation when, in the relatioii connecting two integrals, the function ^ is supposed known. Ex. f). Shew that, if two integrals of the equation are connected by a relation where A, B, C, D are cojistaiits, then Assuming the condition satisfied, integrate the equation. Ex. 6. Two integrals of the equation are connected by a relation of the form Ay^^^By^h,^ + Gy,y^^+Dyi^E=0, where A , B, G, D, E are constants : prove that d^Q .„dQ -z(P-^Gl^q=Q. Shew that the quantity on the left-hand side of this conditional equation ia invaiiantive for change of the independent variable ; and hence, assuming the condition satisfied, shew that the equation can be transformed so as to become a particular case of Lamp's equation (Chap. ix). (Appell.) Equations of the Third Order and Ternariants. 71. Returning now to the differential equation of the third order in the form and supposing that all its integrals are algebraic, we proceed to consider the equation y Google 210 EQUATIONS OF THE THIRD ORDER [71. where F is a homogeneous polynomial in any three linearly independent integrals. For this purpose, it will be convenient to have an equivalent simpler form of the equation which is given by a known transformation*, viz, we have a;+^-»' dt ■ 'iih^-if we take I-- the last of these relations may be replaced by the equation The equation among any three integrals is Consider the simplest case ; it arises when «. = 2, so that F is then a quadratic polynomial involving six terms. Writing a« = a,u, + «iMa + (laMa, where a„ a^, a^ are umbral symbols, the equation can be symbolic- ally represented by We have where u' is du/dt, and so for u". Differentiating again, and replacing u'" by — lu, we have that is, <*„'««" = 0, on using the original equation. Similarly, on differentiating this result, - la^aa- + Ou"^ = 0, that is, «„'■= = 0, " See a paper by tlie author, Fbil. Tram., (1888), p. 441. y Google 71.] AND TERNARIANTS 211 Oil using the first derivative of the original equation. Differen- tiating once more, we have la^au" = 0, 80 that either / = or a^Ou" = 0, If / is not zero, then we must have and therefore, by the second derivative of the original equation, au^ = 0. Hence, on the present liypothesis, we have a^^ — 0, aaa„' — 0, a„^=0, auOu" = 0, au'aH" = 0, oi„"' = 0. Now each of these equations is linear and homogeneous in the six real coefficients that occur in a^^; eliminating these coeffi- cients, we obtain, as equal to zero, a determinant which is the fourth power of M2 . ■ and the latter ought therefore to vanish. But because w,, u^, Ms are linearly independent, this determinant (being the determinant of a fundamental system) dues not vanish — it is a non-zero constant in the present case. Accordingly, the hypothesis that / ie not zero is invalid. Hence i" = ; and therefore, on returning to the original equation, we have Writing our original equation becomes dz^ dz dz ^ Any three linearly independent integrals are connected by a quadratic relation yGoosle 212 TEENABIANTS [71. To obtain the integrals, we note that one value of m is a constant, say unity ; thus where Thus three integrals of the original equation are ^i^, O^O^, 8^, where fi and d^ are two linearly independent integrals of the iatter equation of the second order. It may be noted that three independent integrals of the M-equation ai'e 1, (, f\ so that dt ^ dt , dt I'd,-^- "'S-'' I'di"'- and therefore y^i - Vi'h = 0' thus verifying the existence of the (quadratic relation obtained in a canonical form, Assuming known, we have dz~ 6^' so that and thus three integrals of the original equation are e-. e.l%. ..|/|f. The comparison of these integrals with 6^, Oid^, 6j' is immediate ; for it is a well-known theorem that, if ^i is a solution of an equation g + pe.o, then another solution, which is linearly independent of Sj , is given by Denoting this by 0^, the above three integrals are at once seen to be ft", eA, «.'. y Google 71,] EXAMPLES 213 Ex. 1. Prove tha,t, if a be a solution of the equatioD the primitive can be expressed in the form i/ = Au + Binis.i){a I — J+(7«espf — a |— J, wbere A, B, C are arbitrary constants, and a ia a determiuate constant. What 13 the primitive when a vanishes ? (Math. Trip. Part i, 1895.) Ex. 2. Prove that, if three linearly independent integrals of the equation be connected by a relation F{;y^, y^, 3'3) = 0, where ii™ is a homogeneouiS polynomial of the third degree, then / muist satisfy the equation (567'= - 48/7") 7"'+5477"'«- 1447'/"i'"+ IS^ . 7737'" +^ 24S7'3 - 7 . ZQ^Prr + 84"77'S + ^^^^ /* = 0. Ex. 3. Prove that, if both the fundamental invariants* of an equation of the fourth order vanish, so that it can be taken in the form y -hlOPV +l^P'y +{'il' +97«)./ = 0, then fDur 1 neirlj mdejiendent integrals are given by 8-^, 6-^6.^, 6i6.^\ 6^, where S^ ml fl ire Imeiily indpi>endent intetjrali tf Shew also that, if the relations * These arise in the aame mannec as for the oubio. It the etiuation be transformed by the relations "4 + 40„p + Q,« = 0, sS-^^-=0^ and the fundamental invariants are Os' ^»~^(ii'' ^^ "^ ' p. 210, note. y Google 214 EQUATIONS OF THE THIRD ORDER [71. sulsist among four linearly independent integrals of aa equation of the fourth, order, (so that the integral curve is a twisted cubic), tho equation must he of the above form. Ea:. 4. Construct the equation of the fourth order having fli^i, 5i0ai 6^1, ^2^3 for a set of linearly independent integrals, whore 6^ and fl^, -p^ and ^2, are linearly independent int^rals of the respective equations s+™-". sa+e*-"- Hence infer the form of a quartio equation when a single homogenec quadratic relation subsists among a fundamental system of integrals. Ex. 5. Shew that the equation y"' + ry"+4^' + (6s' + 4r.s)y + 3(s"+!-s')j=0 is satisfied by ;/ = ^, where 8 is an integral of 6"+sB=0; and hence integrate the equation. (Fan Es. 6. Shew that, if five linearly independent integrals of an equation the fifth order are connected by the relations [1 ^1. Vi, Vs, 3'* ||=0. 11 ^2. ys> ^'4. ^i I! the equation can be taken in the form Sj-ao-Sj-iioiSft .:;e-('»£-'-')i-(*S+«'£)»-»^ and thence integrate the equation as far as posisible. (Fano.) 72. Consider now the more general case when three linearly independent integrals of the equation are connected hy an iiTesoluble relation *■&.,</„ y.) = o, where J*' is a homogeneous polynomial of order greater than two : the question is as to the character of the integrals of the equation. For the discussion, it is assumed that the differential equation has its integrals regular and fi'ee from logarithms: it thus is of Fuchsia n type. Let K denote any non- evanescent covariant of the quantic F; such a covariant is the Hessian, which would vanish only if F y Google 72.] AND TEUNARIANTS 215 contained a linear factor. Let z describe any contour, which encloses any one of the singularities, and return to its initial value; the effect upon the fundamental system of integrals iji, y^, yn is to change them into another fundamental system Fi, F,, F3, the two systems being connected by relations Y,= a,y, + ^,y, + rf,y„ (r=l, 2, 3). The determinant of the coefficients a, /3, 7 (say A) is different from zero in every such case ; in the present case, owing to the absence of the term in A^ from the equation, we have {§ 14) A = l, by Poin care's theorem. Now the preceding relations constitute a linear transformation of the variables in the foregoing homogeneous forms ; hence if ^ be the index of K, and ^denote the same function of Fi, Fj, F; as .ff" is of y,, j/g, j/j, we have = K, for fi is necessarily an integer. It thus appears that the value of K is unaltered by the description of the contour. This holds for each of the singularities, as well as for s = qo ; hence K, when expressed as a function of z, is a uniform function. To obtain the form of K in the vicinity of any singularity a, we take account of the fact that the equation is of Fuchsian type : hence in the vicinity we have, for any integral y, (e — a)~py = hoiomorphic function oi z — a, where |^| is a finite quantity. Now K is of finite order in the variables 1/1, y,, y,; accordingly substituting for them, and remem- bering that ^ is a unifoi'ni function of 2, we have {z — a)~''K = hoiomorphic function of « — tt, where <r is an integer, positive or negative. This holds for estch of the singularities, the number of which is limited when Q and R are rational functions of 2 ; it holds also for z=ci:> . Hence K is not merely a uniform function, but it is a rational function, oi z. y Google 216 APPLICATION OF [72. It therefore follows that every eovariant of the quantic F is a rational function of z, exceptions of course arising in the case when the eovariant in an invariant, so that it is a mere constant. Take then any two covariants, say the Hessian H, and any other, say K : we have where and ijf Eire rational functions of z. These are three algebraical equations to determine y,, y^, y^ in terms of z; and therefore the differential equation is integrable algebraically, a theorem first announced* by Fuchs. A case of exception arises, when the Hessian is a constant : the quantic F is then of the second order so that the case has already been discussed ; the integration of the original equation depends upon the integrals of a linear equation of the second order. As an illustration, consider the equation wheii a fundamental set of integrals is coanected by a homogeneous cubic relation. We assume that the equation is of Fuchsian type. Talcing the cubic in the canonical form, we have I being a constant. The Hessian is a rational function, say 1^(1+8?^); so ir=f (y.'+y,' +y,=) - (1 + 2z=) y,y,j<3 = <j. { 1 + 8;«), and therefore Taking the other symmetric covariantt of the cubic, which also ia a rational function, we have and * is equal to a rational function ; so that, ta.kiiig account of the above value of ^I'+ya^+^s', we can write Thus jijS, ^^^, yj' arc the roots of ' Acta Math., t. I (1882), p, 830. t Cayley, Coll. Math. Papers, t. si, p. 345, y Google 72.] COVABUNTS 217 an irreducible cubic. So far aa the coefficients are concerned, they are known to be rational functions of z ; the denominator of each such function is known, because its faotora arise through the aingularitiee of the equation and the multiplicity of any factor can be determined through the associated iadiciaJ equation ; and the degree of the numerator has an upper limit, determined by the behaviour of the integrals for large values of z. Heace and ■^ can be regarded aa known, save aa to a polynomial numerator in each case. We have Tl"' = Afi^ + Bir, + a^ ] the last three being obtained, after diftereatiation, by repeated use of the cubic equation for ij, and the quantities A, B, G, ... being functions of </i, i/' and their derivatives Now writing y = i^ in the differential equation, we find When the above values are substituted and the result is reduced by means of the cubic equation, so that no power of ij higher than the second occurs, we have an equation of the form where Y,, Yj, Y, involve 0, ijr and their derivatives, and are linear in Q, R. As the cubic is irreducible, so that this equation holds for each root, we have Yi = 0, Ya^O, Y3 = 0, three equations to determine and ■^. There consequently exists a relation among the remaining quantities, viz. Q and R : and this must be equivalent to the condition (§ 71, Ex. 2), which must be satisfied in order that the equation .^=0 may exist. Similar results hold for the cubic equation, when the homo- geneous relation between the integrals is of order greater than three ; and corresponding results hold for linear differential equations of higher orders. In fact, if a general homogeneous relation of finite order higher than the second subsists among a fundamental si/stem of integrals of a linear differential equation of order n, then the equation is integrable algebraically: the proof follows the lines of the preceding proof exactly. This range of investigations will not, however, be pursued further, as it becomes mainly formal in character, depending upon y Google 218 APPLICATION OF COVARIANTS [72. the theory of covariants and upon the application of the theory of groups to linear differential equations. An excellent account of what has been achieved, together with many references, is given in a memoir* by Pano who has made many contributions to tho subject ; a memoir-f- by Brioschi contains some investigations con- nected with ternariants; and other detailed references are given in Schlesinger's treatise^, which contains ati ample discussion of the subject. ' Math. Ann., t. Liri (1900), pp. 493—590. + Ann. di Mat., 2" Ser., t. xm (1885), pp. 1~21. J Tkeorie der linearen DiffeTentiatgUiekimgen, ii, 1 (1897), pp. viii — si. The diBCUssioti is to lie found in ohupters 2 — 6 of the tenth seotlon of the treatise. y Google CHAPTER VI. Equations having only some of their Integrals regular NEAR A Singularity. 73. It has been seen that, if all the integrals of an equation are to be regular in the vicinity of each singularity, the coefficients in the equation must be rational functions of z of appropriate form and degree. It may, however, happen that the coefficients are rational functions of s but are not of the appropriate form and degree : in that case, it is not the fact that all the integrals are regular, and it may even be the fact that none of the integrals are regular. This deviation from regularity need not occur at each singularity of the equation : a fundamental system may be entirely regular in the vicinity of one (or more than one) of the singularities, and may not possess its entirely regular character in the vicinity of some other. The conditions necessary and sufficient to secure that all the integrals are regular in the vicinity of a singularity a have already (Ch. Hi) been obtained. If these conditions are not satisfied, then the composition of the fundamental system in the vicinity of the singularity a is no longer of an entirely regular character; we desii'e to know the deviations from regularity. It may also happen that not all the coefficients are rational functions of z; in that case, if uniform, they are transcendental functions and possess at least one essential singularity, say c. Further, owing either to a possibly excessive degree of the numerator in a rational meromorpbie coefficient or to a possibility that z—<x: is an essential singularity of some one or more of the coefficients, it can happen that the conditions for regularity of integrals near a-^oc are not satisfied. The fundamental system y Google 220 EQUATIONS HAVING [73. is then not entirely regular near c or for large values of \z\, in the respective cases indicated, and it may even be devoid of any regular element; the same question as to its composition arises as in the corresponding hypothesis for the singularity a. Accordingly, for our present purpose we assume that the coetBcients in the differential equation are everywhere uniform: that (unless as otherwise stated) they may have any number of poles, and that they may have one or more essential singularities. When tt is a pole of one (or more than one) of the coefficients, and is not an essential singularity of any of them, we have one of the cases just indicated; when qo is a pole of coefficients, not being an essential singularity of any one of them, we have another. We write 1 in these respective cases ; and then our differential equation takes the form d^w d^-'w d'^'^w dvj where the point a; = is a pole of some (and it may be of all) the coefficients. If a!I the integrals were regular in the vicinity of ic = 0, then x'^p^ for r = 1, 2, ..., m would be a uniform function of X that does not become infinite when x=0. As some of the integrals are to be not regular in the vicinity of x = Q, the multiplicity of the origin as a pole of p, must be greater than r, for some value or values of r. Let p^=a:-'^'Pr{s>), (r=l, ..., m), where m-, is a positive integer (which may be zero for particular coefficients), and Pr (*') is a uniform function of a: which does not become infinite when a:=0: also it will be assumed that, unless pr vanishes identically, ■=r, has been chosen so that Pr{0) does not vanish, so that ct, measures the multiplicity of the pole of p, at the origin. Then one or more than one of the quantities w,.-r (r = l,...,m) is a positive integer greater than zero. As in § 23, let r 1 ,,.... yGoosle 73.] ONLY SOME INTEGRALS REGULAR 221 and suppose that ■^lA + 0A-li + 0A-5i>+ ... + 0it*~' + 0o£* is ao integral of the equation, regular ill the vicinity of a; = and belonging to an exponent fj. ; then it is known (§§ 25 — 28) that <^„ is a regular integral also belonging to the exponent fs,, so that where <!>„ is a uniform function of x which does not vanish when x = 0. As this expression, when substituted for w, should make the equation satisfied identically, the aggregate coefHcient of the lowest power of x must vanish (as, of course, must all the other aggregate coefficients). The lowest power of ic in the respective terms has for its index /j.~m, ij. — ^i—{m — i), fi. — 'a^ — {'in — 2), ..., /j. — ia^^i — 1, /j. — 'nr^: and for any other integral, belonging to an exponent <7, the corresponding numbers would be a — m., (r-i!Ti — (m ~1), 17 ~ t^s — (m — 2), ..., <r — OTm_i— 1, <r — st™. Let w. + (m-s)=rj„ (s = 0, 1, ...,'m), and consider the set of integers n„, n., .... n^. Of these, let the greatest be chosen. It may occur several times in the set ; when this is the case, let the first occurrence be at n„, as we pass in the order of increasing subscripts, so that n^<n„ , for r = 0, 1, ...,n-\, n„^n^., r = 0, 1, ....m-n. Then n is called* the characteristic index of the equation : when K = 0, all the integrals are regular. The lowest power of x after substitution of the expression for the regular integral has (U. — n„ for its index ; it arises through p„ — y- — and later terms in the differential equation ; as the coefficient of this- lowest power must vanish, the exponent fi must * Thome, Ci-elle, t. lksv (1873), p. 267. y Google 222 CHARACTERISTIC INDEX AND INDICIAL EQUATION [73. satisfy an algebraic equation of degree m — n. Similarly for an exponent o- to which any other regular ititegi'al belongs ; it also is a root of the same algebraic equation; and each such exponent satisfies that same algebraic equation of degree m — n, which accordingly ia called the indicial equation. But it must not be assumed (and, in fact, it is not necessarily the ease when n > 0) that the number of regular integrals is equal to the degree of the indicial equation. It is clear that, in all cases where n.>0, the degree of the indicial equation is less than in. 71. Suppose now that the given differential equation of order ■m has a number s of regular integrals, which are linearly inde- pendent of one another, where s <m: (the case s = m has already been discussed) : and that there do not exist more than s linearly independent integi-als. After the earlier discussion of fundamental systems, it is clear that any regular integral of the equation is expressible as a homogeneous linear combination of the s integrals, with constant coefficients ; also that, if every regular integral of the equation is expressible as such a combination of s (and not fewer than s) such integrals, the number of regular integrals linearly independent of one another is s. Further, a linear relation among the integrals of the equation, involving a number of regular integrals and only a single one that is not of the regular type, cannot exist ; for the single non-regular integral would involve an unlimited number of negative powers of w, while each of the others occurring in the linear relation involves only a limited number of such negative powers. A linear relation might exist among the integrals of the equation, involving a number of regular integrals and two Integrals that are not of the regular type. We then regard the relation as shewing that the deviation from regularity is the same for the two integrals : and in constituting the fundamental system for the equation, we could use the relation as enabling us to reject one of the non-regular integrals, because it is linearly expressible in terms of integrals already retained. So also for a linear relation with constant coefficients between regular integrals and more than two integrals of a non-regular type. Again, suppose that our differential equation of order m has an aggregate of n integrals, regular in the vicinity of ic = and y Google 74.] THEORY OF BEDUOIBILITY 223 linearly iniiependent of one another; and let it be formed of sub- groups of integrals of the type for'K — 0,l,2,...,ic, where f-A.■v^^,lZ'■-' + ^^l,,i^ Then, after ^ 25 — 28, we know that these n linearly independent integrals constitute a fundamental system for a linear differential equation of order n, the coefficients of which are functions of ic, uniform in the vicinity of a^ = ; let it bo r^^^ '^'d^'^ + i'j(^ =0- Now this equation, being of order n, cannot have more than n linearly independent integrals i and its fundamental system in the vicinity of gs — O is composed of the n regular integrals of the original equation. Hence, by § 31, we must have r^ = a>-''B^(w), {fi = l, 2, ..., n), where Il^{ic) is a holoraorphic function of x in the vicinity of ic = 0, such that R^(0) is not infinite. Accordingly, the aggregate of the n linearly independent regular integrals of the original equation are the n integrals in a fundamental system, of a linear equation of order n of the foregoing type. Eeducibility Of Equationk. 75. If therefore some (but not all) of the integrals of the given equation of order m are of the regular type, it has integrals in common with an equation of lower order. On the analogy of rational algebraic equations, which possess roots satisfying an algebraic equation of the same rational fonn and of lower degree, the differential equation is said to be reducible. Consider two equations ) d™ '■y *(!/)-«. *Sj y Google 224 BEDUOIBILITY OF [75. ■where m>n; and take an expression where the coefficients fio, .B,, ..., Ri are at our disposal, and Let these disposable coefficients bo chosen, so as to make the order of the equation JJ(y)-iliT(!,)! = as low as possible. By taking the ^ + 1 relations P. = B.Q,. -p,=-R,a+-B.(ia'+a), P, = ftft +B,{(i-i) q; + Q,] + B. lii (i - 1) Q." + iq; + a), i'i-ae.+-Bi-.(a'+ft)+-Ki-.(a"+2Q.'+Q.)+..., which determine R^, ■..,Ri, we can secure that the terms involving derivatives of y of order higher than n — \ disappear. Accordingly, writing where S^, S,, ..., Sj are determinate quantities and we have where K is of order less than if. Moreover, if P„, ..., P^, Qo, ■■■, Qn are uniform functions of x, having x = either an ordinary point or only a pole, the same holds of the coefficients R and the coefficients S ; so that L and K are of the same generic character as M and N. From this result several conclusions can be drawn, I. Any integral, common to the equations Jf = 0, N = 0, is an integral of the equation K = 0. If, therefore, every integral of JV = is also an integral of M=0, it follows that K=0 must possess n linearly independent integrals ; as its order is less than n, the equation is evanescent, and we then have «{s).ifiV(S,)). y Google 75.] AN EQUATION 225 II. Any integral, common to the equations JV=0, K~0, is an integral of the equation Jf = ; and therefore, in connection with the first part of the preceding result, the integrals common to M^O, N = constitute the integrals common to i^'" = 0, K=0. The process of obtaining the integrals (if any), common to two given equatioDS ilf =0 and i\r = 0, can thus be made a kind of generalisation of the process of obtaining the greatest common measure of two given poiynomiais. Proceeding as above, we have M =iiV +K \ K = L,K, +K,r where K^, K.2, ..., Kg are of successively decreasing orders. Then unless an evanescent quantity K of non-zero order is reached, sooner or later a quantity K is reached which is of order zero, that is, contains no derivative. In the former case, let ^,+1 be evanescent ; then the integrals of the equation Kt= constitute the aggregate of integrals common to M = 0,N=0. In the latter case, let Kg be the quantity of order zero ; then the integrals common to M=0, ^ — are integrals of lf, = y/W-0. Now f(z) is not zero, for otherwise ^s would be evanescent ; and therefore we have y = 0, the trivial solution common to all homogeneous linear equations. We then say that Jf = 0, A'" = have no common integral. III. An equation having regular integrals is reducible. For one such integral exists in the form y— :•/("). where |^| is finite, andy(fl;) is holomorphic in the vicinity of «= 0, while /(O) ia not zero. We have ydw X f{x) yGoosle 226 CHARACTERISTIC FUNCTION [75, where R(x) is a holomorphic function in t.he vicinity of w = 0, such that R{0) is not zero. Thus the given differential C(^iiation has an integral satisfying the equation that is, it has an integral common with an equation, which is of the first order and is of the same form as itself: in other words, the equation is reducible. But it is not to be inferred that such equations are the only reducible equations. IV. If an equation M = has p (and not more than p) linearly independent regular integrals, it can be expressed in the form where N is of order p, and L is of order m — p. For the p regular integrals are known (§§ 25—28, 74) to satisfy an equation of the form of order p. Every integral of iV"=0 is an integral of M=0; whence, by I., the result follows. 76. We proceed to utilise the last result in order to obtain some conclusions as regards the regular integrals (if any) of a given equation, say, ,^ , , d'"w! d'^^'w dw The result of substituting x^ for w in P (w), where p is a constant quantity, is this is called* the characteristic /unction of the equation 7' = or of the operator P. We have ... +p,^-,^+Pv,; * Frobeaius, Crelle, t. i.ssx (lS7a), p. 318, y Google 76.] INDICIAL EQUATION 227 when the right-hand side is expanded in ascending powers of x, it contains (owing to the form of the coefficients p) only a limited number of powers with negative indices. The highest powers of oT^, arising out of the m + 1 terms in i>;~''P(W), have exponents m, OT,+m-J, OT2 + m-2, ..., ct™_, + 1, vr^, Ho, n., ..., n^. Let n be the characteristic index of the equation, so that n„ is the greatest integer in the set : if several of the quantities 11 be equal to this greatest integer, then n^ is the first that occurs as we proceed through the set from left to right. Denoting the value of n„ by g, let ''^"ffl-^"?'* (*')"?'•' ^'" ~ ^' "' ■■■' ^)' so that 5„ (0) is not zero, and no one of the quantities q^ (0) is infinite. Then «-.P(i')_^-.G(p.«;), where (? is a polynomial in p and is hoiomorphic in a: in the vicinity of a; = 0. Moreover, expanding G(p, x) in ascending powers of x, we have Gip,^)=9,{p)+a:grip) + ..., where each of the coefficients ^ is a polynomial in p, of degree not higher than m; the degree of ffoip) is m — n, and the degree of gg^mip) is ™- Also, g^ip) is the quantity called (§ 39) the indicial function ; the equation ft(f>) = is called the indicial equation. Now take N(w)^a:ffP(w) d^w ^ , d'"-"''!!! dw - 1-^ -£~ + *' £?-T + ■ ■ ■ + «-'^ S + *-'"• where q„ = xs~'^ ; the equation P — can manifestly be replaced by the equivalent J{»)-0, which is taken to be the normal form for the present purpose. We have arPN{af) = G{p,x) = g,{p)-\-xg,{.p) + --, yGoosle 228 NORMAL FORM OF EQUATION [76. which thus contains only positive powers of x when the equation is in its normal form, and which has the indicial function for the term independent of x. We have seen that, if P {w) = possess regular integrals, it is a reducible equation : and the operator P can then be repre- sented as a product of operators. Consider, more generally in the first instance, two operators A and B, each in its normal form ; and let G, also an operator, denote AB. Further, let the characteristic functions of A, B, G, respectively be A (^) = a^^fia:, /,) = «^ 2 /. (p) ^^ = 2 /, (p) a:^+- B{a^) = a^g(x,p)=^x<' 2 g^ip)x'^= 2 g^{p)x'^+>- 0(a:'') = xi'h{x,p)^af 2 h^(p)x''== 2 k^(p}x^+<-, where the summations in f(x, p) and g{x, p) include no negative powers of fl^, because A and B are in their normal forms. Now, as C= AB, we have G{x'-) = AB{af) ^A[%j,{p)ar+'-] = %^l^gAp)/>.(^+p)x^+'^+^ and therefore S 4, (p) «' - ^X ^S J7, (p)/, (M + p) «="'. As X and p. are incapable of negative values, there are no negative values for a- ; and therefore G is in a normal farm. Also fh(p) = gi,{p)f<,{p), 80 that the indicial function of G is the product of the indicial functions of its component operators : and h,(p)= S^y„ (/>)/„_„ (/i + p). Further, if C be known to possess a component factor B which, when operated upon by A, produces C, then A can be obtained. For, take B and C in their normal forms : the equation S h,(p)i^~ t Xsr,(p)A(^ + p)a^+. y Google 76.] CHAEACTERISTIC INDEX 229 then holds. The values of X are clearly 0, 1, ..., so that A is then in its normal form ; and the successive quantities fy are given by the equation for o- = 0, 1, ..., p, the values obtained being polynomials in p, because G is known to be composite of A and B. Of course, this merely gives the characteristic function of the operator ; but the characteristic function uniquely determines the operator. For let /(ic, p) be a function, which is a polynomial in p, and the coefficients of which are functions of x: and let the degree of the' polynomial be m. Then we have* where, taking finite differences in tlie form 4/(«.p) =/('»,(' + !)-/(«,/>), we have <.!«. = 14-/(«,P)1.... Tlius which is the characteristic function of the operator da?'* "* ' dai"-' . + v^x -J- + u„: the operator is determined by the characteristic function. Characteristic Index, and Number of Regular Integrals. 77. Now let the equation of order m, taken in its normal form, be „, , d^""w ,d"'-hjj dw „ i\r (^) . ^^- _- + 5,.-. ^^^ + . .. + 5™-,«^ ^ + 3™«. = ; and suppose that it possesses s (and not more than s) regular integrals, linearly independent of one another. These s integrals y Google 230 CHARACTERISTIC INDEX AND [77. are a fundamental ayatem of an equation, of order s and of Fuchs- ian type ; when this equation is taken in its normal form, let it be ^ ' d^ oaf-' da> where cr,, ctsi -■■. <^a ^^^ holomorphic functions of x in the vicinity of x = 0. As all the integrals of jS' = are possessed by iV=0, there exists a differential operator T of order m — s, such that because N and S are in their normal forms, T also is in its normal form, so that we can take (M'"~' ax'" ' ' ax where t,, T3, ..., Tm-s are holomorphic functions of ic in the vicinity offl!=0. If then Tixfy^xPBix.p), the indicial function of 2' is the coefficient of x" in $ (x, p), which is a polynomial in p and coutains no negative powers of x. This coefficient may be independent of p ; in that case, the character- istic index of 2" is m — 5. Or it may be a polynomial in p, say of degree k in p, where k^0\ the characteristic index of T then is i N' = TS, the indicial function of N is the product of the indicial functions of T and S; so that the ifidicial /unction of S, which gives all the regtdar integrals of N, is a factor of the indicial function of the original equation. The degree of the indicial function of 8 is equal to s, because S = is an equation of order s of Fuchsian type ; the degree of the indicial function of N is m— w, where n is the characteristic index of ^=0. Hence so that (assuming for the moment that k may be either zero or greater than zero) an upper limit for the number of regular integrals which an equation can possess is given by y Google 77.} THE NUMBER OF REGULAR I^TEGRALS 231 where m is the order of the equation, and n is its characteristic index (supposed to be greater than zero). It is known that, when n=0, the number of regular integrals is equal to m. Corollary I. An equation, whose indicial function is a constant, so that its indicial equation has no roots, has no regular integrals; for its characteristic index is equal to its order. But such equations are noC the only equations devoid of regular integrals. CoROLLABY II. When k is equal to zero, then s is equal to m — n, so that the number of regular integrals of the equation is actually equal to the degree of the indicial function. The necessary and sufficient condition for this result is that the equation, which is reducible, must be capable of expression in the form where the indicial function of ?■ is a constant, and tho degree of the indicial function of S is equal to the order of S. This result, which is of the nature of a descriptive condition, appears to have been first given in this form by Floquet*. Other forms, of a similar kind, had been given earlier by Thom^f and by FrobeniusJ (see | 83, post). Note. On the basis of the preceding analysis, it is easy to frame an independent verification that the characteristic index is not greater than m — s. For in the operator T, the quantity Tm^s^ic does not vanish when a: = ; and all the quantities r>, , such that \<m~ s — k, do vanish when a: = 0. Hence, when we take iV as expressed in the form the coefficient of is the first (in the succession from left to right) in which Tm„(_i occurs; it also contains q^, t,, .... Tm-s-s-i, all of them occurring * Ana. de I'^o. Nona. Sup., 2° S6r., t, vui (1879), Buppl, pp. 63, 64. t Crelle, t. lxxvi (1S73), p. 286, t GrelU, t. isns (1875), pp. 331, 832. y Google 232 NUMBER OF REGULAB INTEGRALS [77. linearly. When a;=0, all of these except t™„j_j: vaoish, and Tn-t-n does not vanish ; and therefore qm-a-i does not vanish when a!=0. In the coefficient of of- -, — , where iJ.>s-\-k, the quantities i/o, ti, ..., t^_„ occur linearly: each of these vanishes when ic = 0, and therefore jm-» does vanish when ic = 0. As this holds for all values of /i, it follows that (jms-k is the first of the quantities q which does not vanish when ic=0; hence the characteristic index of N is in — b — k, that is, it is ^m—s, where s is the number of regular integrals possessed by the equation iV = 0. E<c. 1. If w=Mii be an integral, regular and free from l<^arithm8, of an equation P=0, which is of order m and haa i regular integrak, and if a new dependent variable u be given by ahew that u satisfies an equation Q = 0, which is of order m - 1 and has s — 1 regular integrals ; and obtaiQ the relation between the characteriatic index of P=0 and that of §=0. (Thomd.) Ex. 2. The equation s integrals regular in the vicinity of - = and linearly independent of th d=0 lltjJ P hwthtt pl(t 1 1 t ) f ea h f th m g uefii t p (Tl m ) b ti^ ly as gn d bj t t th d t th t =0 pi d yptp thttl m wffi i p be d t n ed a. t perm tth qt tp ss — bt lyaasged gila, t gr 1 h ealj d p d t f th (rhm^) ^4P thtth dt ydflitt tht luftt \=0 f i d 1 vmg d If t i d f, yhllh yilealydpdt-^l t 1 thtV lllbepltfthfm QMD wh-eth d If i la i Q M D are of degrees 5, 0, m-y-h respectively, and D is of order m-y-S. Is there any limitation upon the order of Mf (Cayley.) Ex. 5. Shew that an equation QD=0 has at least as many regular integrals as Z> = 0, and not more than Q = and 5 = together ; and that, if all the integrals of D=0 are regular, then QI>=0 lias as many regular integrals as §=0 and i)=0 together. y Google 71.] AKD DEGREE OF INDICIAL FUNCTION 233 Hence (or otherwise) shew that, if an equation F=0 has all its integrals regular, then F can be resolved into a product of operators, each of the first order and such that, equated to zero, it has a regular integral. la this resolution unique? (Frobenius.) 78. In the two extreme cases, first, where the degree of the indicial function is equal to the order of the equation, and second, where its degree is zero, the number of regular integrals is equal to that degree. The preceding proposition shews that, in the intermediate cases, the degree merely gives an upper limit for the number of regular integrals. It is natural to enquire whether the number can fall below that upper limit. As a matter of fact, it is possible* to construct equations, the number of whose regular integrals is less than the degree of the indicial function. Taking only the simplest ease leading to equa- tions of the second order, consider the two equations U = ^ + ky + h = 0, F=J + % = 0, da! " dx " of the first order ; and form the equation ax ax which manifestly is of the second order, say d'y dy If we can arrange so that ic = is a pole of p of order n, where n ^ 2, then « = in general will be a pole of q of order ;i + 1 ; and the indicial function will then be of the first degree. Consider now the equation of the second order. Since 0"- V^-h, it can be written which is satisfied by i — - dx where A is any ai'bitrary constant. V (1872), pp. 311—313. y Google 234 THOMfi's THEOHEM [78. Let Y be an integral of the equation of the second order. It may be an integral of F = ; if it is not, then, when we take that is, 1/, is an integral of U= 0. Thus any integral of the equation of the second order either is an integral of 1^= or is a constant multiple of an integral of ^7=0. If, then, U=0 and V=0 are such that they possess no regular integral, the differ- ential equation of the second order can possess no regular integral ; at the same time, its indicial function is of the first degree. The equation V"= will not have a regular integral, if a; = is a pole of k of order greater than unity ; and the equation U=0 will then not have a regular integral, if /i is a rational function of x. Ex. 1. The aggregate of conditions can be aatiisfled aimnltaneously in many ways. For instance, take '-s+i. «- Z+^x -s-i; The differential equation of the second order is d^y 1_ c^_3 + 2^_, dx' ^ dx its indicial equation is of tlie first degree, and it has no regiilar integrals : or the number of its regular integrals is less than the degree of its indicial equation. The conclusion can otherwise be verified ; for it is easy to obtain two linearly independent integrals in the form 1-7^ no linear combination of which gives rise to a regular integraL Ex. 2. Shew that the equation has no regular integrals : and verify the result by obtaining the integrals of the equation, (ThomiS, Floquet.) y Google 79.] DETEBMINATIOS OF THE REGULAR INTEGRALS Determination of such Regular Integrals- as exist. 79. When the degree of the indicial function of an equation of order m is less than m, no precise information is given as to the number of regular integrals possessed by the equation. The further conditions, sufficient to determine whether a regular integral should or should not be associated with any root of the indicial equation, can be obtained in a form, which is mainly descriptive for the equation of general order and can be rendered completely explicit for any particular given equation. Let the equation be A {w) = go^'" ^^ + q,x"^' -^^^ + . . . + <;™w = 0, of characteristic index n. Let E{0)he the indicial function , and let a be one of its zeros, so that Then, if a regular integral is to be associated with a-, it n)ust be of the form w = a^(c„+ c,« + Csa;^+ ... + CpX^ + ...). This expression, when substituted in the equation, must satisfy it identically, so that, after substitution, the coefficient of «"+*■ must vanish for every value of p : and therefore where the number of terms in this difference -relation depends upon the actual forms of g,,, q,, ..., 5™. Of the coefficients /|„/i, .■■,fr, the first is which is of degree m — n in p; of the remainder, one at least, viz. fg-m, is of degree m in p, where g has the same significance as in §76. The successive use of this difference -relation, together with the equations for the earlier coefficients, the first of which is leads to the values of all the quantities (;^h-c„, for the successive values of fj. ; and thus a formal expression for it is obtained that y Google 23C DETERMINATION OF THE [79. satisfies the equation. If, however, the expression is an infinite series, it haa no functional significance when it diverges : that this frequently, even generally, is the case, may be inferred as follows. For if c^+i -i- c^, with indefinite increase of (i, tends to a limit that is not infinite, so also would C,i-|-a-i-c„+i, C/^^^c,^^, and so on; and therefore for finite values of o, also would tend to a limit that is not infinite. Now a number of the quantities 7.W' for various values of 6, undoubtedly tend to zero as jj, increases indefinitely ; some of them may have a finite limit ; but one at least is infinite, viz. /.w ■ because the numerator is of degree n higher than the denominator, both of them being polynomials in /i. Consequently, the ex- pression acquires an infinite value as i* increases without limit. The difference -relation requires the value of the expression to be always - 1, so that the hypothesis leading to the wrong inference must be untenable. Therefore c,^i-i- c^, with indefinite increase of /i, does not tend to a limit that is finite, and therefore the series diverges*. There is then no regular integral to be asso- ciated with the root a. * It is not inconceivable that, for special values of in and of n, and for special forms of the eoefEoients q, as well as for a speoial value of the limit c„4.j4-Cji, the infinite parts of the expression ,J,/7W "•, might (lisappear, and the espression itself be eiiual to -1. In that case, the series would oonvei^e : and an exception to the general theorem would occur. But it is dear that such an eioeption is of a yery special character: it will be left without further attempt to state the conditions explicitly. y Google 79,] REGULAR INTEGRAIS THAT EXIST 237 As the series thus generally diverges when it contains an un- limited number of terms, the regular integral is thus generally illusory. The only alternative is that the series should contain a limited number of terms : and then the regular integral would certainly exist. Accordingly, let it be supposed that the series contains k + 1 terms, so that Ci Ci Ct are quantities known from the difference-relation, and that Ck+i, c*+2. ■■■ ad inf. all vanish. If we secure that cj+i, ct+i, ..., Cn+r all vanish, then every succeeding coefficient must vanish in virtue of the diffei'ence- relation ; and these t relations will then secure the existence of a regular integral to be associated with the exponent cr. Taking p — k, k~l, ..., k — r + 1 in succession, we find the t necessary conditions to be /„ (k) Ci = 0, that is, /„ (k) = 0, and generally for values r= 1, 2, ..., t- 1. The first of these is /S(a- + k) = 0. so that the indicial equation, which possesses a root er, must possess also a root <T + k, where k is a positive integer. (In the special instance, when k = 0, no condition is thus imposed : in the general instance, when & is a positive integer greater than zero, it is easy to verify that E{a- + k) is the indicial function for ic = x .) When the aggregate of conditions, which will not be examined in further detail, is satisfied in connection with a root of the indicial equation, a regular integral exists, belonging to that root as its exponent; and there are as many regular integrals, thus determined, as there are sets of conditions satisfied for each root of the indicial equation. Explicit expressions for the various coeOicients c can be derived, when the explicit forms of the quantities q are known : but the general results involve merely laborious calculation, and would hardly be used in any particular case. The results are therefore. y Google 238 MODE OF OBTAINING [79. as already remarked, mainly descriptive : and so, in any particular case, it remains chiefly a matter for experimental trial {to be completed) whether a regular integral ia necessarily i with a root of the indicial equation. For this purpose, and also for the purpose of c regular integrals associated with a multiple root of the indicial equation, a convenient plan is to adopt the process given by Frobenius (Chap. Ill) when all the integrals are regular. We substitute an expression W = CoClf -It CiiC^+'+ ... +c^af-^i^+ ... in the equation of characteristic index n. After the substitution, the ttrst term is where E{p') is the indicial function, of degree in — n; and we make all the succeeding terms vanish, by choosing the relations among the constants c appropriate for the purpose. We thus have N{w) = cE{p)a^; and the relations among the constants c are of the form where the constants ix^^^-i' ■■■> '^co ^^'"^ polynomials in /i and, when this relation is the general difference- relation between the coeiBcients c, one at least of these polynomials a^r is of degree m in II. When the difference -relation is used for successive values of fj,, we obtain expressions for the successive coefheients c, which give each of them as a multiple of c^ by a quantity that is a rational function of fi. When these coefficients are used, we have the formal expression of a quantity vj which satisfies the equation Unfortunately for the establish men t of the regular integrals, this formal expression does not necessarily (nor even generally) con- verge: for, in the difference-relation among the constants c, the right-hand side is a polynomial of degree m in [i, while the left- hand side is a polynomial of degree m — m in /i, so that the series Se^a^''''' would, as in the preceding investigation, generally diverge. y Google 79.] EEOULAR INTEGRALS 239 But while this is the fact in general, it may happen that the series would converge when p acquires a value occurring as a root of the equation £(p) = 0. In that case, the series satisfies the equation if(»)-0; in other words, it is a regular integral of the differential equation. Further, if the particular value of p be a multiple root of the indicial equation, it can happen that the series div Tp converges for this particular value of p ; and then ^g).l|o.«(p).-) = 0, because the value of p is a multiple root of E= ; in other words, p- is then a regular integral of the differential equation. And so possibly for higher derivatives with regard to p, according to the multiplicity of the root of E=0. The whole test in this method is therefore as to whether the series converges for the particular value (or values) of p given as the roots of the indicial equation. The method of dealing with a repeated root of the indicial equation has been briefly indicated. Corresponding considerations arise, when £" = has a group of roots differing among one another by integers. In fact, all the processes adopted (in Ch. iii) when all the integrals are regular, are applicable token only some of them are regular, provided the various series, whether original or derived, are converging series. The deficiency, that arises through the occurrence of divei^ing series, represents the deficiency in the number of regular integrals below m — n. As already stated, the tests necessary and sufficient to discriminate between the convergence and divergence of the various series are not given in any explicit form, that admits of immediate application. y Google 240 EXAMPLES [79. £x. 1. Consider the equation constructed in § 78, Ex, 1. The indieial equation is p-3=0, BO that there ia not more than one regular integral ; if it esists, it belongs to an exponent 3. To determine the existence, we substitute in the original equation ; that it may he satisfied, we must have 0=^{(» + 2)(™ + l)-2}c„_, + nc„, for « = 1, 2, .... We at once find and therefore The series S c„a-^*" diverge^, and tlierefoie the one possible le^ulii lute^iil does not exist; that is, the oiigmal equation po^isc'ises no leguUi intPoHl, although the indieial equation is of the first degiee If there were a regular integral, it would satisfy an equation where w is a holomorpbic function of ^ ; and the original equation could then be written (''£-) ('i-')-. where » is some holonsorphic function in the vicinity of x-=0. It might be imagined that, as the indieial equation is of d^ree unity (a property that does not forbid the existence of a regular integral), it would be possible to obtain the regular integral through a determination of u, and that the divergence of the series in the preceding analysis is due to the operator which annihilates only expressions that are not regular. That this is not the case may easily bo seen. We have so that, if the resolution be possible, we have y Google 79.] EXAMPLES 241 Substituting in the second of these the value of v given by the first, we find as an equation to determine «, supposed a holomorphio fuiiotion of x. Let >>e substituted ; in order that the equation for v, may be satisfied, we have and, for values of m higher than zero, (K + 2ao-l)ii, + a(a^a,_i+asa,._s+...)-a„ + ,=0. Hence Oo=3, a,=4, a^=2i, and so oo. The relation giving «„+„ when taken for successive values of n, shews that all the coefficients a are positive ; hence >(« + 5)<i„, that is, 30a„>(™ + 4)!, and so the series for m diverges : in otber words, there is no function m, and the hypothetical resolution of the equation is not possible. Mole. This ai^umont is general ; it does not depend upon the particular coefficients for the special equation that has been discussed. Sx. 2. Consider the equation which is in the normal form. The characteristic index is 1 ; the indicial equation is (9(fl-l)-5d+9=0, {fl- 3)2=0, BO that the number of regular integrals cannot be greater than two, and such as exist belong to the exponent 3. To detei'mine these regular integrals (if any), we adopt the Frobenius method of Ch. Iii, Taking ^=c„xP + c,afi*'^ + ... + c^sf + " + ..., provided _ p2-2p-5 ■^1 '''* p-2 ' and, for values of n greater than unity, = (p + «.-3)c,+{p^ + p(2n-4) + )i2-4H-2}c„.i--2(p + n)c„_a, a factor p + m — 3 having been removed, because it does not vanish for these values of ?i. Let (p+»-3)e,-2«„_, = 4„. yGoosle 242 DETERMINATION OF REGULAR INTEGRAL [79. X;. = (p-2)<!,-3o„=-(p-3)(p + l)c„. Also the difference-equation for the coetKeients c becomes i:,=l^l)'->{p + n)ip + n~l)...{p + 2)k, =.(_!)» (p_3){p + l)(^ + 2)...{p + ™)c„ Hence, writing 2" """nO. + ft-S)'*'' in the rtslation and substituting the value of i,„ we have Adding the aides of tllis equation, talsen snccessiveiy for m, b-1 3, 2, and noting tliat .J{5 + »p-i>")n()>-3)t„ ...,.[M5+2,-,')n(p-3) + (,-3)J_(-i)."Jsi»yiisi"^]. We thus have a value of y in the form where S (-irn(p + m)n(p+m-4) :-^2"-.(5 + .p-p^) 'V;^-- +(p-3)^" n(p-3) _ _ nMn(p+»-a) and this satisfies the relation DV=c„{p~3fxi: It is clear that formal solutions of the original differential equation ai '^- [f],. Of these, the first is in effect, a constant multiple of sfl^ ; and the second is y Google 79.] WHEN IT EXISTS 243 because a series, in which is the coefficient of j^*^, manifestly diverges*. It thus appears that, although the indicial equation for x=0 is of the second degree, the differeutial equation possesaes only one integral which is regular in that vicinity ; and this integral is a constant multiple of x^e^. This regular integral satisfies the equation «*-(3+»),.0, SO that the original equation must be reducible. It is easy to verify that it can be expressed in the form , + .|-(3 + 2.)}{.*-(3 + 8.);,}.0. JH.v, 3. As an example which allows the convergence of the series for the regular integral to occiur in a diffei'ent way, consider the equation ^y'_(l_2a;4-2a^)y+(I-2jr;+;i;2)y=0. The indicia! equation is P=0, so that one regular integral may exist. To determine whether this is so or not, we substitute which (if it exists) belongs to the exponent aero. Comparing coefficients, and, for all values of n that aro greatei' than unity, (B + I)«„^.i = (»^ + m + l)o„-3aa„_i + a„_j. Let In general, the values of a (and the consequent values of a) as determined by the last equation, lead to diverging series ; but in our particular case, o that 0^=0, 0^ = 0, and generally Cm^O, that is, • The series in ijg is saved from divergence because, in it, these ooeffioieiits are muldplioii by the taotor p - 3, which vanishes for the special value of p and whidi therefore removes the quantities that cause the divergence in the Eecond. integral. y Google 244 and therefofe EXISTENCE OF [79. provided so that a regular integral esiats. It is a constant multiple of e'. Ex. 4. Consider the equation i) (J.) = (aH5 4- ^k") y"" + (^ + 4iK«+ i!*) y'" - (2ii72 + 3j!»+ ar«) y + (3ar + toH 4iJ^) y - (3 + to + 4a;S) ,?/ = 0, The characteristic index is unity ; hence the numher of regular integrals is not greater than three. To determine them, if they exist, we take an cxpi'eBsion and form 2)(y), choosing relations among the coefficients c such that all terras after the first in the quantity D (y) vanish. We thus find Cip2(p-2) + c„(p-l)(p-2)(f,Hp-3) = 0, and, for values of n greater than unity, t.(p + »-ir'(j. + »-3) + <.-,(p + — 2)(p+«-3)& + »)'-(p+«)-3) + «.-,(|, + l>-S)'(p + »-4)(p + »)_0. The indkial equation is (p-l)'(p-3).0 of degree 3 aa was to be expected ( = 4 — 1), be use tl is 1, The roots form a single group; ifaregulir t o the root 3, it wiii be free from logarithms ; f tw eg 1 belonging to the root 1, one of them may or may t b fr t and the other will certainly involve logarithms. Consider the root p=3. As p+K-3 thei nish f values of n, wo may remove it from the differen qu fc <^„(p+«-l)'' + o.^,(p + »-2){(p + ™)=-(p + «)-3} c,(p + m-l)Hc,_,{p + m-2)(p + »-3) = *„, we at once find i„ + (p + )i)i„_i = 0. We require the value of Jc^. We have, for p = 3. til gt i- m 1 g thm J th t th 1 tt Taldng *a = 1602 + 6e, = 80Co. i„ = (-l)''(« + 3)(»i + 2)...6ij = §(~l)"(«.-i-3)!<r,; yGoosle 79.] REGULAR INTEGRALS ao that, writing Aa «| and «a are positive, it follows that all the and clearly 80 that the series divei^s ; and there iis no regular integral belonging to the root 3. Moreover, the coefficient of o„, being (p+n—l)\ does not vanish when p=3 for any value of TO ; hence, if two regular integrals exist belonging to the root unity of the indicial equation, one of them will certainly be free from logarithms. Consider now the repeated toot p = l. As p + )i — 3 vaniahes for this value of jj when «.=2, the difference-equation is then evanescent for ji = 2 and it does not determine r^. For other values of n, the quantity p4-«-3 does not then vanish, so that it may be removed. We then have, for values of m ^ 3, the same form of equation as before, viz, ■^„(p + m-I)^ + c„-i(p4-ft-3){(p + »t)^-(p + ™)-3} + c„_,(p-f»-3)(p + «-4)(p+«) = 0. Also c,= -(p-I)(pS + p-3)^, the value p = l not yet being inserted because we have to differentiate with regard to p. Tlie difference-equation for jj=3 gives 9(3-1-1802 = 0, For values of n.^4, let p = iT-2, so that the value of o- is 3 ; take m-2=m, ao that the values of m are ^ 2 ; and write then the difference-equation becomes 6,„(^+m-l)'-f&™-.(<r + m-2){(<r-t-w)^-(ff + i«)-3! -H6«_2(o-t-m-3)(o-Hm-4)(o-l-m) = 0. Here <r=3, m^2; e2=&o, C5 = 6[--: exactly the same as in the former case ,^[b^-a^x+a % - ) with tl o cailier notation it certainly diverges inles^ 6j|=0. If 6=0, every ciefecient vanishes, and the senes itself vanishes As we require regular mte^rils we shall therefore assume 6^ = that is, i.;=0; and then all the remaining c effiuenta \inisli so that we h \e 1= .[■^- ■'"^'(p-l)(p'+p-3)^], yGoosle 246 EXAMPLES [79. au osprossion which is such that are integrals of the equation The former ia o^s : one regidar integral thus is The latter ia another regular integral in The original differential equation accordingly has two regular integrals. S!x. 6. Shew that the equation haa one integral regular in the vicinity of :i;=0-, and express the equation in a reducible form. Sx. 6. Shew that the equation has two regular integrals in the vicinity of x=0, in the form e", x^ ; and obtain the integral that is not regtilar. Ex. 7. Shew that the equation has no integral, that ia regular in the vicinity of m=<i ; ej[preas the equation in a reducible form, and thence obtain the integral by quadratures. (Cayley.) Ex. 8. An equation P — can be e.^pressed in the form $0 = 0, where ^=0 has no regular integrals ; can P=0 have any regular integrals ? Illustrate by a special case. Ex. 9. Ill the equation the coefficients P are poljDomials in a: of degree p, and p<.n: shew that it possesses n—f integrals, which are integral fuuctions of a;. (Poincar^.) y Google lUKElJUCIELE EQUATIONS Existence oe laREUUCiBLE Equations. 80, We have seen that an equation is reducible when it ia satisiied by one or more of the integrals of an equation of lower order, in particular, by the integral of an equation of the first order. The main use so far made of this property has been in association with the regular integrals of the equation: but it applies equally if the equation possesses non-regular integrals that satisfy an equation of lower order. It is superfluous to indicate examples. It must not be assumed, however, that every equation is reducible by another, if only that other be chosen sufiiciently general. On the contrary, it is possible to construct an irre- ducible equation of any order m, as follows*. We construct an appropriate characteristic function which, as is known (§ 76), uniquely determines the equation. Take a poly- nomial in p of degree m, say let the coefficients of the powers of p be holomorphic functions of «, not all vanishing when « = 0; and let the function, subject to these limitations, be so chosen that, when arranged in powers of X in the form h {^., p) = h, (p) + xh, (p) + a;'h,(p} +..., ho (p) is independent of p and not zero, and h^ (p) is of degree m in p. Then if iV=0 is the equation determined by h(x, p) as its characteristic function, JV= is irreducible. Were N reducible, an equation S = of lower order s would exist such that each of its integrals satisfies N = <i; and then an operator Q, of order m — s, could be found such that N=QD. Wc take Q and D in their normal form ; and so N is in its normal form. Now Q{af) = af {^,(p) -(- a^{p) +,af^,ip) + ...[, yGoosle 248 REDUCIBILITY OF EQUATIONS [80. the right-hand sides of which are polynomials in p of degrees m — s and s respectively. Then, as in § 76, we have K ip) = ?„ ip) vi (p) + ?. (p) v« ip + 1)- Now /(„ (p) is a constant, being independent of p ; hence, owing to the polynomial character of Q {w") and D (a^) in terms of p, the two quantities ^((p) and ijoCp) £"^e constants. Accordingly, ijo{p + 1) is a constant ; and therefore the degree of f.</>)i.(p)+f.(p)i.((>+i) in p is the degree of 171 (p) or fi (p), whichever is the greater. But the degree of i)i(p) is not greater than s, and that of ^i{p) is not greater than m — s; so that, as s > 0, the degree is certainly less than m. But the espresaion is equal to A, (p), which is of degree tn. Hence the hypothesis adopted is untenable ; and the equation N=0, as constructed, is irreducible. Equations having Regular Integrals are Rgducible. 81, Suppose now that, by the preceding processes or by some equivalent process, the regular integrals of the equation N=0 have been obtained, s in number, and that the equation of which they constitute a fundamental system is S = 0, of order s: a question arises as to the other m — s integrals of a fundamental system of N= 0. Let where T and S (and therefore also iV) are taken in their normal forms. The s regular integrals of if, say y,, y^, ..., ijs, all satisfy iS = ; and no one of the m—s non-regular integrals of N, say W], Wj, ..,, Wm—s. satisfies jS = 0, for this equation has all its integrals regular. Let S(»,) = »„ (.-=1 m-,); then, as N{wr) = 0, we have r(«,) = o. Now Wr is not a regular expression; hence n^ is not regular, that is, it contains an unlimited number of positive and negative y Google 81.] HAVING HEGULAK INTEGKALS 249 exponents when it is expressed as a power-series. Accordingly, the m — s quantities u are integrals of the equation which is of order m — s and has no regular integrals ; and the m~s non-regTilar integrals of iV= are given by S (»,) = «,. it being sufficient for this purpose to take the particular integral and not the complete primitive of the latter equation. The case which is next in simplicity to those already discussed arises when s = ni — 1, so that the original equation then possesses only one integral which is not regular. The equation T^O is then of the first order. With the limitations laid down, the normal form of T is d where q, and q, do not become infinite when a; = 0. As the integral of T(u) = is not regular, it follows that ^i does not vanish and that g,, does vanish when a:= 0; so that, if where a is a positive integer > 1 and Q {x) is a holomorphic function in the vicinity of « = 0, sach that Q (0) is not zero, the equation determining u is 1 ^ a. A_ _9i - say lg + ^i + ?i<^ + ... + ? + iiW-o, where iJ (a:) is a holomorphic function of x in the vicinity of iC = 0. This gives %+ -»i, + -.. + --' u-ar-e' ' -P.W, where Pi is a holomorphic function of x in the vicinity of a: = ; and then to determine w, the non-regular integral of N—0, we need only take the particular integral of yGoosle 250 REDUCJBTLITY [81. where in which qa, qi, ..., Jm-i denote lioloraorphic functions of x, and ^o does not vanish. Writing the equation for v takes the form where qo, p,, p^, ..., pwt-i are holomorphic functions of «, such that 5o and pm-i do not vanish when iC = 0. In some cases it happens that a particular integral of this equation exists, in the form of a converging power-series repre- sented by «'— "— -PW. where P{x) is a holomorphic function of a;: in each such case, the non-regular integral of the original equation is «■"'— «°-PW. But, in general, the particular integral of the u-equation is not of the same type as the regular integrals of the original equation : and then the non-regular integral of the preceding equation cannot be declared to be of that type. Ex. An illuatration is furniahed by the equation in Bs, 6, § 79, viK. It has two regular integraJa, viz. and these constitute the fundamental system of y Google 81.] THE ADJOINT EQUATION 23 in the normal fonn. To have the given equation in the normal form, v multiply throughout hy x'' \ and then it must be the same as when f is properly determined. We easily find that and so the equation for determining m, where y being the non-regular integral, is v.<ix~ 3?{\■\■'i^•\■%a?^^a^) ao that Hence the non-regular integral of the original equation <ai'i particular integral of „ ^ , 1-(-2j; + 3j;3 + :k* \ f-y^y^— -^ 1". Let j/ = v^ ; the equation for v ia easily found to be "(-i) satisfied by v = l : and therefore the non-regular The Adjoint Equation, and its Properties. 82. Of the properties characteristic of a linear equation, not a few are expressed by reference to the properties of an associated equation, frequently called Lagrange's adjoint equation. It is a consequence of the formal theory of our subject, as distinct from the functional theory to which the present exposition is mainly limited, that Lagrange's is only one of a number of covariantive equations associated with the original. As its properties have been studied, while those of the others remain largely undeveloped. y Google 252 PROPERTIES OF lagrange's [82. there may be an advantage in giving some indication of a few of its relations to the original linear equation. The latter is taken in the customary form P(?^) = P,wi»' +P,wi"-'» + P,w"'-=' + ... +P„«; = 0, where w'*"' is the rth. derivative of w with respect to z; and from among the various definitions of the adjoint equation, we choose that which defines it to be the relation satisfied by a quantity v in order that vP(w) may he a perfect differential. Now, on inte- grating by parts, we find + ^, (vPr) W l"-^^> -...+(- 1 )"-^ [w ^- {vPr) dz, for all the values of r\ hence, writing P. =/-•., R (w, v) = p„w '""" 4- piW "'~^' + . . . + pa-iW, PW = P,.-^(P,_„) + ;|(P,.-..)^.., + (-1)"£(.P.), we have jvP (w) dz = R (w, v) + iwp {v) dz, and therefore .P(«.)-«p(.).^(K(»,.)}. It is clear that, in order to make vP{w) a perfect differential, whatever be the value of w, it is necessary and sufficient that v should satisfy y Google 82.] ADJOINT EQUATION a linear equation of order n, commonly called Lagrange's adjoint equation ; and further that, if v is regarded as known, then a first integral of the equation P (w) ;= is given hy B {«,.«) = .. a being an arbitrary constant, and R being a function manifestly linear in w and its derivatives. Further, since jwp (v) d^^-R (w, V) + jvP (w) dz, it is clear that wp {v) is a perfect differential if P(») = o, shewing that the original equation is the adjoint of the Lagrangian derived equation: or the two equations are reciprocally adjoint to one another. Ex. Shew that, if ic,, ,.., w„ be a fundamental system of integrals of the et[uation i*(w) = 0, then a fundamental system of int^rals of the, adjoint equation p{v) = lS is given by 1 -Ir." Shew also that the product of the respective determinants of the two sets of fundamental integrals depends only upon 1'^. One immediate corollary can be inferred from the general result, in the case when the equation P{w) = Q is reducible. Suppose that p(«,)=p.p,(»)=p,(fl'), say, where W^P^iw); then we have (vP(w)ds= Lp,(W)dz ^R,(W,v)+jwP,(_v)dz, ■where P, is the adjoint of P„ and R^ is of order in W and in v one unit less than P,. Again, writing F=P,W, y Google 254 THE ADJOINT EQUATION [82, we have where P^ is the adjoint of P^, and iE^ ia of order in V and in v one unit less than P,. Combining these results, we have jvP(w)dz^R,(W,v)-\-R,{V,v) + jwP,(V)dz = R{w,v) + jwP,P,(v)dz, where R is of order one unit less than P in w and in v. It follows that P,P,{v)^0 is the adjoint of P(»)- p.p. (") = !), where P, , Pi are adjoiut to one another, and likewise Pj, Pj. By repeated application of this result, we see that the adjoint of P<«/j=P,P,...P,(M.) = is given by P;P^,...P,P,(i')=0, Hence the adjoint of a composite equation is compounded of the adjoints of the factors taken in the reverse order. Manifestly an equation and its adjoint are reducible together, or irreducible together. The expression R (w, v) is linear in the derivatives of w, up to order n — 1 inclusive, and also in those of v, up to the same order : it may be called the bilinear concomitant* of the two mutually adjoint equations. For further formal developments in respect to adjoint equa- tions and the significance of the bilinear concomitant, reference may be made to Frobeniusf, Halphonj, Diui§, Cels|], and Darbouxt. * Begleitender hilinearer Differentialaitsdmclc. with Frobeniua, t CreUe, t. lxixv (167S), pp. 1S5— 213; referencea ace given to other writers. t Liouville'a Journal, i' S^c, t. i (1885), pp. 11—85, § Aim. di Mat., 3» Ser., t. n (1899), pp. 297—324, ib., t. ni (1899), pp. I2S— 183. II Ann. de VEc. Norm., 3' Sir., t. viii (1891), pp, 341^15. IT Theorie generale dee surfaces, t. ii, pp. 99 — 121. y Google 82.] EXAMPLES 255 Ex. 1. Prove that, if ii linear equation of the second order ia self-adjoint, it is cspressible in the form that if a linear equation of the third order, in the form ia effectively the same as its adjoint equation, then and find the conditions that a self-adjoint. r equation of the fourth order should be Ex. 3, Prove that, if the equations yoHW-i-nyiVi""'!-!- "Yi — y.j»ff"~^4----=0, e adjoi it to one another, then ri= -ffi+ffo'y ^1= -7i + To'' 73= -ffs + ^ffi-^ffi"+go"\ ?3= -■ V3 + 3y2■-■V + 7o"'> and obtain the expression of the bilinear concomitant. i„ denote any « arbitrary functions of x, such that dx^-'-' does not vanish identically; and suppose that these functions of x are regular in a given region of the variable, aa well as the coefficients a of the etjuation Further, let a set of quantities p bo constructed according to the law dpn dp, dv„_, »-"•• ?■-"<-£. »-'".-j;f' ■■•■ '••-'•---^nr- y Google 256 PROPERTIES OF laorange's [82, and let the last of them be denoted by - Z, so that there are n functions Z con'osponding to the n functiona z. Shew that, if Q (c) is the value of § when the last column of the latter iiS replaced by constants c„ ..., c„, if §{a', a\) is its value when the last column is similarly replaced by Sj (a:,), %(a:i), ..., j„ (x^, and if Q {x, a^i) is its value when the last column is similarly replaced by 2i (:Kj), iTj (^cj), ...,Z„(a;,), then where a is a value of 3! within the given ri^ion and the constants c are determined in association with a. Indicate the form of this result when z^, ..., 2„ are a fundamental system of the equation, which ia the adjoint of the left-hand side of the above equation. Also shew how, in even the most genera! case, it can be used as a formula ! to obtain an infinite converging series of integrals as an 1 for y. (Dini.) 83. Consider an expression Piw) and its Lagrangian adjoint p (v), and let R (w, v) denote their bilinear concomitant ; then ,P(»)-»;,(rt.^j(2i(»,„)), which holds for all values of v and w. Accordingly, let w = 2~''~''~^, V — z", where s ia any integer ; then ^p (,-,—.) _ ,-,-.-,^ (^) _ ^ ^M(z->—'. ^-)). Now the left-hand side is a series of powers of z, having integers for indices ; as it is equal to the right-hand side, which is the first derivative of a similar series of powers, the left-hand side must be devoid of a term in s"'. Let Piz% =SMt)z^^-, be the characteristic function of P {w) ; then the coefficient of z~^ in scP {2-^-'-') is /s (- p - 5 - 1). Further, let be the characteristic function oip{v); then the coefficient of 3~' in s"*"-^'^ (s**) is 4>»{p)- Hence >^,{(>)=M-p-s-n yGoosle 83.] ADJOINT EQUATION 257 and therefore so that, if be the characteristic function of a given equation, then S/, (-(>-/" -1)2"' is the characteristic function of the adjoint equation. When P{w) is in its normal form, all the coefficients /^(p) vanish for negative values of fi, but /i,(p) is not zero. Hence f^{— p — fi — l) vanishes for negative values of p,, but not /ti(~ p ~^)'} and therefore the adjoint expression p(v) is in its normal form. Moreover, their indicial functions y, (p), <p^ (p) are such that /o(/>)=0o(-p-i), Mp)=A(-p-n 80 that they are of the same degree*, or the characteristic indices are the same. Hence if an equation has all its integrals regular in the vicinity of a singularity, the adjoint equation also has all its integrals regular in the vicinity of that singularity ; for the characteristic index is then zero for the original equation, and it therefore is zero for the adjoint equation. Similarly, if am equation has all its integrals non-regular in the vicinity of a singularity, tlie adjcnnt equation also has all its integrals non- regular in the vicinity of that singularity ; for the characteristic iudes: is then equal to the order of the original equation, and it therefore is equal to the (same) order of the adjoint equation. On the basis of these two results, we can obtain a descriptive condition necessary and sufficient to secure that, if a differential equation of order m has an indicial function of degree m~n, the number of its regular integrals is actually equal to m — n. Let P=0 be the differential equation, with an indicial func- tion of degree m — n. Let R=0 be the differential equation of order m — n, which has the aggregate of regular integrals of P = for its fundamental system ; its indicial function is of degree ra-n. Then (§ 75, iv) the equation P = can be < the form * Thom^, Crelle, t. lssv (1873), p. 276; Frobeiiiua, Crelk, t y Google 258 Lagrange's [83. where Q is a differential operator of order n. Because the degrees of the indicial functions of P and li are equal to one another, it follows (from § 76) that the degree of the indicial function of Q is zero, that is, the indicial function of Q is a constant, and therefore (§ 77, Cor. i) the equation Q = has no regiilar integral. Now construct the equations which are adjoint to P = 0, Q = 0, R — respectively ; and denote them hy p = 0, ? = 0, r = 0, Because R and r are adjoint, and hecause all the integrals of R = are regular, it follows that all the integrals of r = are regular; and conversely. Similarly, because Q and q are adjoint, and because Q = has no regular integral, it follows that q = has no regular integral ; and conversely. Further, by § 82, we have p^rq, so that the equation adjoint to P = is p = rq = 0, and this equation possesses all the integrals of 5 = 0, an equation whose indicial function is a constant. Hence it is necessary that the equation adjoint to P = should possess all the integrals of an equation of order n, having a constant for its indicial function, if P = is to have m — n linearly independent regular integrals. But this descriptive condition is also sufficient to secure this result. For, as the condition is satisiied, we have p = rq, where the indicial function of q is 2. constant ; hence, with the preceding notation, we also have P^QR, and the indicial function of Q is a constant. Accordingly, as the indicia! function of P is of degree m — m, it follows (§ 76) that the indicial function of R is of degree m — n; and therefore (Ch. Ill), as the order of R — is m — n, all its integrals are regular. But P = possesses all the integrals of P = ; and therefore it has m~n regular integrals. We therefore infer the theorem : — In order that an equation of order m, having an indicial function of degree m — n, may possess m — n regular integrals, it is necessary y Google 83.] ADJOINT EQUATION 259 and suffia'piit that the adjoint equation should possess all the inte- grals of an equation of order n, having an indicia! .function which is a consUvnt This result was first established by Fiobenius*; and it may be compared with the corresponding result obtained by Floquet (§ 77). The special case, when w = 1, had been previously discussed by Thom^f, who obtained the result that an eqtuition of order m, having am, vndiciat function of degree m — 1, possesses m — 1 regular integrals, if the adjoint equation has an integral of the form /©_!».«-, where G(-] is a polynomial in - , and a is a constant. We shall not pursue this part of the formal theory of linear differential equations further : we refer students to the authorities already (§ 82) quoted, as well as to Thom^j, Floquetg, and Griinfeldil. * Grdle, t. lxks (1875). pp. 331, 332. t Crelle, t. lisv (1873), pp. 278, S79. X A summary of many of the memoirs a Thomfi, published in CreJie's Journal, will t pp. 185—281. % Ann. de VEc. Norm. Sup., 2' Ser.. t. viii (1879), Supplement, p. 132. II CTelle, t. oxY (1895), pp. 328—842, ib.. t. oxvii (1897), pp. 278-290, ib., i. cxiii (19tX)), pp. 43-52, 88. y Google CHAPTER VII. Normal Integrals; Subnormal Integkai^. 84. It is now necessary to consider those integrals of the differential equation in the vicinity of a singularity, which are not of the regular type. Suppose that such an integral, or a set of such integrals, is associated with a root 9 of the fundamental equation (§ 13) of the singularity which, as in the last chapter, will be transformed to the origin by the substitution 1 z — a = x, z — -, according as it is in the finite part of the plane, or at infinity. Let p denote any one of the values of then it is known that an integral exists in the form where i^ is a uniform function of x in the vicinity of the origin. As this integral is not of the regular type, the function ^ will contain an unlimited number of negative powers, so that the origin is an essential singularity of ^ : in the case of the integrals con- sidered earlier, the origin was either a pole or an ordinary point. Accordingly, when ^ is expressed as a power-series, it will contain an unlimited number of negative powers: it may contain an unlimited number of positive powers also, and in that case it has the form of a Laurent series. Classification of such integrals might be effected in accordance with a classification of essential singularities ; but the discrimina- y Google 84.] NORMAL INTEGRALS 261 bion that thus far bas been effected among essential singularities is of a descriptive type *, and has not led to functions whose general expressions are characteristic of various classes of singularities. Accordingly, it ^ possible to choose one function after another with differing forms of essential singularity, and to construct (where practicable) the corresponding linear equations possessing integrals with the respective types of singularity : but there is no guarantee that such a process will lead to a complete enumeration. There is one such function, however, which is simpler than any other, and yet is general of its class. It suffices for the complete integration of the linear equation of the first order when the origin is a pole of the coetScient ; and an indication has been given (§ 81) that it may serve for the expression of an integral of an equation higher than the first. The equation of the first order may be taken to be where a^+'P is a holomorphic function, s being some positive integer. Let where /' (ic) is a holomorphic function ; then we easily have where "^ {x) is a holomorphic function of w, and n = 2+i-i+ •■• + -■ a!* a^ ' X It is clear that ic=0 is an essential singularity of the integral; and also that we thus have the complete primitive of the equation of the first order. It appeared, in § 81 and the example there discussed, that such an expression, if not in general, still in particular cases, can be an integral of an equation of higher order. As all expressions of the form y Google 2(i-2 THOMfi'S NORMAL [84. where il is a polynomial in - , possess the same generic type of essential singularity, we proceed to the consideration of equations that may possess integrals of this form. Such an integral is called* a normal elementary integral or (where no confusion will occur) simply normal. The quantity e", through the occurrence of which the point a) = is an essential singularity, is called the determining factor of the integral; the other part of the integral, being ie<'iy {x) where i^ is holomorphic, is of the type of a regular integral, and so the quantity p is called the eivponent of the integral. Construction of Normal Integrals. 85. We proceed, in the fii'st place, to indicate Thome's method f of obtaining snch normal integrals as the equation d™ti! d"^^-!/) dw _ die™ '^ dx™~^ ' ' ' "™~^ ^(c "™ may possess. (The method gives no criteria as to the actual existence of normal integrals: and therefore, if any criteria are to be obtained for equations of order higher than the lirst, they must be investigated otherwise.) If a normal integral exists, it is of the form where ii is a polynomial in - ; and il is determined so that, if possible, the equation satisfied by u may possess at least one regular integral. Lot ftc" so that (,= 1, *, = n', tp^,^tp' + Q.'tp, {p=l,2, ...); then d"w _ /, , du , d"~'u ^ d"M\ dx" \ die dx" ' daf-J * Thom^, Crelle, t. sov (1883), p. 75. Cajley, ib., t. c (18S7), p. 286, suggeated the came suin-egular ; but the name normal is that nhioh has geiierall; been adopted. t Crelte, t. lsxvi (1873), p. 292. y Google 85.] INTEGRALS 263 When these quantities, for the successive values of n, are substi- tuted in the differential equation for w, the determining factor e" can be removed ; and the differential equation for w then is d-t. d—'u du , , (".-D! „, , ("'-2)> , ■)! ' (r-l)!(m-i ■)!"'"' (r-2)!(>n-i-)!' ...+(»t-r + l)p. forr = l, 2, ...,m. If the origioal equation possesses a normal integral, then, after the proper determination of fl, the differential equation for u will possess at least one regular integral : its characteristic index cannot then be greater than m — l, which (after the results in the preceding chapter) is a necessary but not a sufficient condition. As 12 is a polynomial in «-', its form and degree being un- known, let its degree be s — 1, so that s^2; we then have for £1' an expression of the form il' . «£ , «3 , Hence in ti, the governing term (that is, the term with highest negative exponent of a:) is -^ ; in ij, it is -J^ ; and so on, so that, in *„, it is -^. As in § 73, let ot, denote the multiplicity of « = as a pole of p, ; then in q^, the governing exponents of its respective parts are rs, ^, + (r-l)s, ■=r, + (r-2)s, ..., ^r-i + s, ^,. Thus the governing exponents in q,. are, so far as they go, less than those in q^+i by s, and s ^ 2. Hence, in forming the characteristic index for the equation in u, for the purpose of determining whether it may possess a regular integral, the governing exponent in q^ is certainly greater by s than that in any otiier coefficient; the characteristic index is m, the indicial function is a constant, and the equation has no regular integral. But, thus far, ii is quite arbitrary ; and it may be possible, by proper choice of its constant y Google 264 DETERMINATION OF [85. coefficients, to secure that a number of the terms in q^ with the greatest exponents of *■"' shall disappear. If by thus utilising the governing exponent and the constants in SI', we can secure that the characteristic inde\ of the equation in u is less than m, the indicial function cea-ses to be i con^tint and the equation may have a regular intejfral In Older that the indicial function may not be a constant, the governing exponent of q , must be less than that of q^, by unity at the utmost, or that of q,^ must be less than that of q^ by two at the utmost, or (tor sjme value of }) the governing exponent of qm-r must be less than that of 5 by ? it the utmost ; whereas at the present moment, these diminutions are s, 2s, rs respectively, where s ^ 2. Hence an initial necessity is that the s — 1 terms in qm with the highest exponents of a:'^ shall vanish. Now qm,^tm + Pltm-l + ■■■ +Pm~-A+l^m- The s — 1 terms in t^ with the highest exponents ol' le"' are the same as in II'", because of the form of il' and because (but not more than those s — 1 terras are the same) ; hence the s~l terms with the highest exponents of (e~^, say the first s — 1 terms, in n'"' +piii''"~' + . . . +p„-jfi' +p„ must vanish. 86. To render this result attainable, it is necessary that the greatest exponent must not occur in only a single term of the preceding expression, for then the term could vanish only by having d, = ; the greatest exponent must occur in at least two terms. Consequently no one of the numbers ms, ■=T, + (m — l)s, ■5r2 + (m~2)8, ..., ot„, may be greater than all the rest, that is, no one of the numbers 0, ^i~s, W3 — 2s, .... vT^ — ms, may be greater than all the rest. Of the quantities y Google 86.] NORMAL INTEGRALS 265 let g be the greatest. Evidently g is greater than unity ; for the original differential equation has not all its integrals regular, and so CT„ > w for at least one value of n. Now s cannot be greater than g\ for any such value would make all the integers in the series 0, ^i — s, «;,— 2s, ..., Wm-ms, negative except the first, that is, the first would then be greater than all the rest. Hence s^g: and g ^ 2, from the nature of the I. When g <2,no value of s is possible; and then there is no normal integral of the type indicated. Such a case arises for the equation when p and q are holomorphic in the domain of ic = and neither vanishes when a; = 0. The quantity g is the greater of 1, |, that is, it is less than 2 ; so that there is no normal integral. Moreover, as the indicial function of the particular equation is a constant, it has no regular integral. II. When g is an integer (necessarily greater than unity), wc manifestly might take s = ^. For two at least of the numbers would then be equal to the greatest among them, which is zero ; and then two at least of the numbers ms, JiTj + (m ~ 1) S, Wa + (jrt — 2)s, ..., ■nr^_i + S, CT^, would be equal to the greatest among them, one of these being ms. More generally, let n be the characteristic index of the original equation, so that for all values of /t that are greater than w; then, adding {m—n)(«—l) to each side of the inequality, we have ■OT„ + (m - n) s^w^ + im-ii^s + ifi- n)(s - 1), where fi>ii. In the case of all these numbers, {/j. — n){s—l) is certainly positive ; so that the first s — 1 terms in our expression y Google 266 NOEMAL INTEGEALS [86. are not affected by the quantities coiTesponding to -a^i^ + im — )j.)s, and they can occur only through the quantities corresponding to w^ + (m - X) s, for X= 0, 1, ...,n, where T>r„ = 0, and n is the characteristic index of the original equation. We thus consider the first s ~ 1 terms in and this holds for any value of s equal to or greater than two. As regards g, which is the greatest among the quantities it occurs only among the first n, in the present circumstances ; for it certainly is greater than unity and if any one of the last m — n, (say — liT^ is the greatest of these last m — n), is greater than unity, then because we have n /J- \li /\n 1 that is, for /i is greater than n. Thus g does not occur in the last m — ii of the quantities, if one or more than one of them is greater than unity ; and it certainly does not occur among them, if no one of them is greater than unity. Hence g is the greatest among the quantities It may occur several times in this set ; let - tct, be the first occur- rence, in passing from left to right, and - ^^ be the last. Take first s = g\ then we have ■^^■\-{wi, — k) h = mg, OT,. •\-{rn, — r)s = vig, ■wx + (m - \) s < mg, if \ < k, or if X > r ; y Google 8(5,] THEIR DETERMINING FACTOR 267 SO that the highest terms of ail, being those with index Trig, occur in ii'", p,ii'"-~ + ... +prn.'«-': K then p, = x''^-{c, + d„a; + ...). (,7 = 1, 2, ...), the equation which determines a^, the coefficient of a:~i' in il', is o/ + c^a/-' + . . . + c,. = 0. The remaining g — 2 coefficients in il' are given by equating to zero the coefficients of the next ^ — 2 terms in fl'" + piil'"-' + . . . + p„_iii' -I- p„. Each set of values of the coefficients determines a possible form of Xi' and therefore a possible form of determining factor. The number of sets, different from one another, is ^ r. The preceding cases arise through s = g; but if g, being an integer, is greater than 2, other values of s, less than g, may be admissible. They can be selected as follows*. Mark the points 0, n; -a-,, n-l; wj, n - 2 ; ...; «r„, 0; in a plane referred to two rectangular axes; and taking a line through the first of them parallel to the axis of a;, make it swing round that point in a clockwise direction, until it meets one or more of the other points ; then make it swing in the same direc- tion round the last of these, until it meets one or more of the remaining points ; and so on, until the line passes through the last of the points. There thus will be obtained a broken Hue, outside which none of the marked points can lie. If a line be drawn through any of the points, say sj,, n — v, at an inclination tan"';t to the negative direction of the axis of ^, its distance from the origin is (i+^r'K+(»-«>*'l, so that, for a given direction /a, the distance is proportional to It therefore follows that an appropriate value of s is given by any portion of the broken line, which is inclined at an angle tan~"'/i to ' Thy method is due to Fuiseux; see T. K, S 96. y Google NORMAL INTEGRALS the negative direction of the axis of y, where ^ i integer, ^ 2 : the value of s being As many values of s are admissible as there are portions of the broken line with inclinations tan~^/(., where /4 is a positive integer, which is > 2. For each admissible value of s, arising from a portion of the broken hne, the terms in which correspond to the points on that portion, give the tertns of highest negative power in x. If, for instance, a portion of line, having as its extremities the points corresponding to p,Q'"~' and p(il'"~', {t>r), gives a value g' (necessai-ily an integer, as being a value of s), then the coefficient a^- satisfies an equation Cra/~'' + ... + c, = 0, and the remaining ^ — 2 coefficients in li' are obtained in the same manner as before. Each set of values of the coefficients determines a form of li' and therefore also a possible determining factor; and the number of sets different from one another is ^t — r. And so on, with each piece of broken line that provides an admissible value of s. III. When the greatest of the quantities is greater than 2 but is not an integer, we construct a tableau of points as in the preceding case, and draw the corresponding line. Only such values of s (if any) are admissible as arise from portions of the line, which are inclined at an angle tan~^ p. to the negative part of the axis of y, fj. being an integer > 2, 87. In every case, where a possible form of il' and thence a possible form of fl have been obtained, we take y Google 87.] SUBNORMAL INTEGRALS 269 If a normal integral of the original equation exists, the equation for M must possess a regular integral ; and each regular integral of the latter determines a normal integral of the former having the determining factor e". An upper limit to the number of integrals thus obtainable is furnished by the degree of the indicial function of w ; but the investigations of the last chapter shew that, when the degree of the indicial function is less than the order of the differential equation, the number of regular integrals may be less than the degree and might indeed be zero. The simplest mode of settling the matter is to take a series of the appropriate form, determined by the indicial function of the w-equation, substitute it in the differential equation, and decide whether the coefficients thence determined make the series converge. The normal integral exists or is illusory, according as the series convei-ges or diverges. When the normal integral exists, we say that it is of grade equal to the degree of il as a polynomial in ar^. Subnormal Integrals. 88. In the preceding investigation of normal integrals, it was essential that the number s should be an integer ^ 2 : and accordingly, such values of j/., as were given by the Puiseux diagram and did not satisfy the condition, were rejected. But though they are ineligible for the construction of normal integrals, they may be subsidiary to the construction of other integrals. Let /t denote such a quantity, given by the Puiseux diagram in the form of a positive magnitude that is not an integer: its source in the diagram makes it a rational fraction which, being expressed in its lowest terms, may be denoted by h-i-k. The terms which, for this quantity as representing a possible degree for il', have the highest index of (ir' in Il'« + piil'"-^' + ... +p„-,.ri' +p,„ are those which corr^pond to points on the portion of the line that gives the value of /t. Hence, taking y Google 270 SUBNORMAL [88. an equation is obtained by making the aggregate coefficient of this term of highest order disappear ; the equation determines A. Now take a new independent variable f such that and make it the independent variable for the differential equation ; dD. A ao that and therefore h — k ^ Thus ii is infinite when a: = 0, provided h>k, that is, for values of jj. that are greater than unity. Accordingly, when we proceed to consider the differential equation with ^ as the variable, values of fi of the preceding form can be obtained by the earlier method : in fact, we may obtain a normal integral of the equation in its new form, the conditions being that the equation for v. which results from the substitution shall have a regular integral or regular integrals. When once the value of k is known and the transformation from a; to |^ has been effected, the remainder of the investigation is the same as for the construction of normal integrals of the untransforraed equation. Examples will be given later, shewing that such integrals do exist. As they are of a normal type in a variable x", where k is a. positive integer, they may be called subnormal'^. Their existence appears to have been indicated first by Fabryf, 89. We have seen that, if g denote the greatest of the quantities ^x, W^. \^,, ..., * Poiiicare, Acta Math., i, viii, p. 304, ealis them anormaUs. + Sar Us intigrahs du iquations di^'erentieiiea linSaires b, coefficients rationels, jThfise, 1883, Gauthier-ViUttra, Paris), Section iv. y Google 89.] INTEGRALS 271 and if the equation possesses a normal or a subnormal integral of the form then il' is a polynomial in ^~' (or in some root of 3~') of order equal to or less than ff ; and therefore il is a polynomial in s~^ of order equal to or less than g — 1- Let then Jt is called the rank of the differential equation for s—0. When li is an integer, the grade of a normal integral may be equal to M: if not, it is less than B. When R is not an integer, let p denote the integer immediately less than R ; the grade of a normal integral may be equal to p or may be less than p. When k R Ts 3, fraction, equal to -: when in its lowest terms, then a sub- normal integral may exist having a determining factor e", where fl is a polynomial of degree k in z ' ; it will still be said to be of grade -j in s, that is, of grade R. All subnormal integrals are of grade R or of grade less than R. Ex. Olitaiii t)ie rank of the equation for 2 ^ CO , the coefB.oiBiits p being polynomials in £. 90. The converse proposition, due^ to Poincare, is true as follows : — If n normal or subnormal functions are of grade equal to or less than B, and have the origin for an essential singularity, they satisfy a linear differential equation of order n and rank not greater tha R fa Any n f net ns "^at sfj ■i 1 r I tt rentiai equation of order n : in the j sent case let tie ^+P ^ ■+ +PnVJ = 0. a a * Ada Ml 8 p 30 ii s been somewhat altered, so as to admit the □ ma and n a tegrals together. y Google POINCARfi'S THEOREM ON [90. Let the normal and the subnormal functions be arranged in a sequence of descending grade: when so airanged, let them be so that, if S,, i^a, ..., Rnhe their respective grades, R ^ ill ^ -Ra ^ ■ ■ ■ ^ -Rn— ! ^ -Rji— 1 ^ -B«- a the fundamental determinant of the n functions, viz. Now d"-' and A„,y is obtained from A by substituting the derivatives of order n for the derivatives of order Ji — ?• in the rth row. The value of P,. is In order to obtain the degree of s = as an infinity of P,., it will be sufficient to consider only the governing terms in A and A,(^r; and the degree is determined through the differences between the two sets of most important terms in the rth rows. Now if d'iwp = 0. We taku out of where 0g_j, is finite (but not zero) when i the pth column a fector for each of the n values of y; we take out of the mth row a factor for each of the n values of m ; and then every constituent in the surviving determinants A an<i A„,r is finite. The initial terms in y Google 90.] NORMAL AND SUBNOKMAL INTEGRALS 273 these constituents are the same for all the rows except the (r — l)th: the difference there is that ai", Oa", ..., a„" pccur in A'„,ri while ai""', Oj""^, ..., ««""*■ occur in A', where A'„_, and A' are the modified determinants, and Oi, Og, ..., a„ are the coefficients of the governing terms in fi,, fl^. -■-^ ^n- Accordingly, if A' =As^+..., then A'„,r = ^V4-..., where the other indices are higher than $, and A, A' are constants; and therefore i ^ 3-inin-ii (Bi+ii eSOp a^iTp ^.^ the summation in the exponents heing for values I, 2 n of p. Hence Now A, being the fundamental determinant, does not vanish identically : and as ^ = is an essential singularity, and not merely an apparent singularity, A does not vanish when s = 0; thus A is not Jiero. It might happen that ^' = 0; but in any case, if -ST, denote the order of 2 = as a pole of the coefficient P,, we have i^,.Kr(Ri+l}. Thus the largest of the numbers -z^r is <.R,+ 1 ^ R + l ; and therefore, for z = 0, the rank of the equation < R, which proves the proposition. When all the integrals are normal, which is the circumstance contemplated by Poincare, the quantities R are integers and the determinants A', A'„^, are uniform ; so that the coefficients P then are uniform functions of z. The coefficients P are uniform also when the aggregate of subnormal integrals is retained : the proof of this statement is left as an e Note. An equation, which has a number of normal integrals, is reducible; so also is an equation, which has a number of sub- normal integrals. By the preceding proposition, the aggregate of the normal integrals (or of the subnormal integrals) satisfies a linear equa- tion with uniform coefficients, say JV = 0, of which they are a F. IV. 18 yGoosle 27i EXAMPLES OF [90. fundamental system. Denoting the original equation by P = 0, we can prove, exactly as in § 75, that P can be expressed in the form where Q is an appropriate differential operator. In other words, P is reducible. The investigation of the detailed conditions, imposed upon the form of P by the possibility of such reducibility, will not be attempted here. Further, it must not be assumed (and it is not the fact) that retlucible equations ai'e limited to equations, which have regular, or normal, or subnormal integrals. Ex. 1. Consider the equation whore p, q, T are holomorphic functions of a: that do not vanish when ^ = 0. To investigate the possible kinds of determining factor, we form the tableau of points 0, 3 ; 3, 2 ; 5, 1 ; 7, ; and then eonstruGt the broken line. There are two pieces ; one gives ;< = 3, the other fi = 3 ; the former joins the first two points ; the last three lie on the latter. The possible ospreasiona for O' are therefore where a and 3 ^''S uniquely determinate, and y is the root of a quadratic equation. Of course, the actual existence of normal integrals depends upon the actual forms of p, q, r. Ex. 2. Shew that the equation y"'+^?'y"+^ »'+^ '3'='' where p, g, r are holomorphic functions of ,r that do not vanish when si=Qi, possesses no normal integrals in the vicinity of j; = G: but that it may possess subnormal integrals. Ex. 3. Consider the equation which has no regular integral, because the indicial function is a constant. The numbers s'j, org, ctj are 1, 2, 6; so that j — 2, and we therefore take <=3, so that y Google 90.] NORMAL INTEGRALS 275 We have to make the single (s-l) highest power of a.'~' vanish, in the eipansion of in ascending powera of x ; hence so that a is a cube root of unity, and Q = ". Accordingly, we write after reduction, the equation aatisfied by u is found to be ,„ _ 3M"(a-^+6aa^) The indicial equation for x=0 is a'(8-l).0, which has a single root 3 = 1 ; so that the w-equation possibly may possess a single regular integral which, if it esiats, will belong to the exponent 1, and so will bo of the form As a matter of fact, the w-equation is satisfied by as may easily be verified ; and thus the original equation possesses a normal integral where a is a cube-root of unity. But a may be any one of the three cube roots of unity ; and therefore the original equation in y possesses the three normal integrals .^(^+«^), ^i^+a^^^), e^(x+a:,), where <i is now an imaginary cube-root of unity. The singularities of the equation given by l+6i-^ = are only apparent (§45). jEe. 4. Prove that the equation has, in the vicinity of x^a^, two linearly independent normal integrals, provided a. is of the form pip + 1), where ^ is an integer ^0 ; and obtain 18—2 yGoosle 276 NORMAL INTBGEAIS [90. Bx. 5. Prove that each of the equations aY' + ars/ - («s + 2*2) 3^ = 0, has, in the vicinity of ;i;=0, two hoearly independent normal integrals ; and obtain them. Ex. 6. Prove that the equation has, in the vicinity of ^ = co, three linearly independent normal integrals; and obtain them. Ex. 7. Prove that the equation possesses one normal integral in the vicinity of x = ; and that one normal integral is illusory in that vicinity. Ea:. 8. Shew that the equation (x + %)3fiy"'-^{x'-i-Zx-'i)^'-{a:-\-%)xy.--{1ix'^-5x-^)y==Q posaeaaes three normal integrals in the vicinity of x=Q. [They are iee" , xe " , xe "loga^.] Ex. 9. Prove that a solution of the equation is expressed by where n' = a^-ib, m(X + l) = a(,7 + l)^c. (Math.,Trip., Part i, 1896.) Hamburgek's Equations. 91. The conditions, sufficient to secure that an equation, of order in and not of the Fuchsian type, shall have a regular integral, have not been set out in completely explicit form (§1 78, 79) ; and consequently, the conditions sufBcient to secure that such an equation shall have a normal integral have not been set out in explicit form. The foregoing examples (§ 90) afford y Google 91.] hambueger's equations 277 illustrations of the detailed process of settling such questions in individual instances ; and the following investigation* gives the appropriate tests for a particular class of equations, which afford an illustration of the general method of proceeding. We consider the equation „ _ a + Ss + j^ 1 which a must be different from zero (§ 86) if the equation is to s a norma! integral. For any integral that occurs, 3 = is singularity. For large values of ^, the integrals are regular ; and a fundamental system for s= co is composed of two regular integrals, which belong to exponents — p, and — pa arising as roots of the quadratic equation p{p-\} = y. These two regular integrals may be denoted by where Pj, P^ are converging power-series. As the origin is the only other singularity of the equation (and it is an essential singularity), it follows that P, and Pj have a = for an essential singularity ; all other points in the plane are ordinary points for P, and P^. The expression of a uniform function having only a single essential singularity, say the origin^.aud no accidental singularity, is known by Weierstrass's theoremf to be of the form where P (- J is a uniform function having all the zeros of the original function (the simplest form of P being admissible), and ff ( -] is a holomorphic function of - which is finite everywhere except at 2 = 0. * It ia due to Harabui^er, CrelU, t. an (188S), pp. 238—273. + T. F., § 52. y Google 278 SPECIAL EQUATIONS WITH [91. The function g may be polynomial or it may be transcendental ; the discrimination depends upon the character of the origin as an essential singularity for the original function. As the present application is directed towards the determination of normal inte- grals, the function g ( - J will be taken to be a polynomial in - . If the original function has an unlimited number of assigned zeros in the plane outside any small circle round the origin, P is transcendental. When the number of zeros is limited, P[-) is a polynomial in - , which can be taken in the form where A; is a finite positive integer, / is a polynomial in s of degree not greater than k, its degree being actually k when ^ = oo is not a zero. The equations to be considered are those which have integrals ...p,(l), ^p,(i). as above, one (or both) of the functions P, and Pj having only a limited number of zeros outside any small circle round the origin, with the further condition that the essential singularity at the origin is of the preceding type. Thus an integral is to be of the form say, where il is a polynomial in - , the exponent o- is a constant, and /(z) is a polynomial in z; and the differential equation for u is to have a regular integral which, except as to a factor z", is to be a polynomial in z. Let then the equation for u is y Google 91.] NORMAL INTEGRALS 279 After the earlier explanations, it is clear that we must take The equation for u then is " 2a , , 2a- 0- ye --0, which is to have a regular integral of the type u = .'/(,) = z''(c„ + c,z+ ■■■ + C„3" + ...), there being only a limited number of terms on the right-hand side. The indicial equation for s = is - 200- + 2a - /3 = 0, BO that ■2a Substituting the expression for u, and equating coefficients, we have, after a slight reduction, {(n-\-a)(n + ^-l)^y]c, = {2a(n + <r} + ^]c^+, = 2a(n+l)c„+,; and therefore (n + a) (n + <r-l)-'y '"■'+' 2«(h+1) It is clear that, if the series with the coefficients c were to be an infinite series, it would diverge and the integral would be illusory. For this reason also, as well as by the initial condition, all the coefficients from and after some definite one, say after Ck, must vanish ; and therefore we must have or substituting for er its value, we see that the qi.iadratic equation whff)-e o? = a, m.iist have a positive integer {or zero) for a root. This condition is sufficient to secure the significance of the series, and therefore sutScient to secure the existence of a normal integral of the equation " _ « + /3 s + "/g' yGoosle 280 A CLASS OF EQUATIONS [91. Clearly, there are two values of a. If for either value the condition ia satisfied, there is a normal integral of the form where a has .the value for which the condition is satisfied. The condition cannot be satisfied for both values, if the values of <r are different, and arise from different values of p ; for if it could, we should have 2a 2a Now pi + /3i= 1 ; and therefore il + 7^2 = pi - 0-, + p, - (Tj = - 1, which is impossible, as neither ky nor k^ is negative. The condition can be satisfied for both values of a, if the values of (7 are the same, that is, if 13 = 0: for then the condition, that the equation {0 + l)O=j can have a positive integer as a root, shews that the equation „_ a^+ 7^ w -- ^ w possesses two normal integrals of the form e^aCCo + ca+.-.+c^"), e"'s(c,-c,z+...±ce^''). The condition can be satisfied for both vahies of a, if the values of it arise through the same value of p, whether they are the same or not ; and the equation then possesses two normal integrals. The limitations on the constants are given in the first of the succeeding examples. Ew. 1. Prove that the equation w s4 ■"■ possesses two normal integrals, if where q is any integer, positive, u^ative, or zero, and. /i is an integer that may not vanish. (Hamhurger.) y Google 91.] HAVING NORMAL INTEGEALS 281 Ex. 2. Obtain the conditions sufficient to secure that the equation v/' + 2-^vf + [ + 3j + y«M^S^(Z* may have a normal integral of the foregoing type. Can it have two normal integrals ? Eie. 3. Prove that the equation possesses two normal integrals, if a is an integer (positive, negative, or zero). Ex. 4. Prove that the equation ^ ■ jfl possesses a normal integral if the quadratic equation has a positive integer (or zero) for one of its roots for either value of ^a. What happens (i) when both its roots are integers for the same value of ^a, (ii) when, for each value of ^a, the equation has a positive integer for a root ? Ex. 5. Prove that the equation V"'-2ft(n + l)^ + 4™(w + l)^4-|^w(™ + l)(™+3)(»i-2) + <i'}«'=0, where re is an integer and a, is any constant, has four normal integrals of the where ^ ( - ) is a polynomial in - . (Halphen.) 92. In an earlier paper, Cayley* had proceeded in a different manner. If where {z) is a holomorphic function of z not vanishing with z, we have W 3 0(S) • CreiJs, t. c (18S7), pp. 286—295; Coll. Mn.l\. Paperf, vol. sii. pp, 444—152. y Google where R(z) is a hotomorphic function of 2 in the vicinity of the origin. Further, if where (s) is a holomorphic fiinctiou of s not vanishing with s, and li is a polynomial in - , we have W S (s) fJiw, say, where Ii(z) is holomorphic in the vicinity of the origin. Cayley transformed the equation by the substitution and then proceeded to obtain, from the differential equation for y, an expansion in ascending powers of 2. When once a significant expression for y has been obtained, the value of w can immediately be deduced. Applying this method to the equation „ ^ a + /3g + 7^ ° w ^ - w, the equation for y is at once found to be Hamburger's investigation shews that the integrals of the equa- tion in vj are „..,»P,(lj, „,.,»p,(l), which are valid over the whole plane but have ^ = for an essen- tial singularity. If an integral, say w^, has an unlimited number of zeros, the origin being its only essential singularity, then* any circle round the origin, however small, contains an unlimited * I', f.. g§ 32, 33. y Google 92.] METHOD 283 number of these zeros: so that if, in the vicinity of the origin, the expression of w, is ^ (a) would have an unlimited number of zeros within the small circle so drawn. The expression for ^ is but the function ,) ! has an unlimited number of poles in the immediate vicinity of the origin, and so the right-hand side cannot be changed into an expression of the form where m is a finite integer. Accordingly, the assumed expansion is not valid in this case : and the method does not lead to signifi- cant results. But when the integral has only a limited number of zeros, so that ^{z) is expressible in the form in the vicinity of s = 0, where ^ f - j is a polynomial in - and /(s) is a polynomial in a that doe; be changed into an expansion is a polynomial in a that does not vanish with z, then "^ can and so the assumed expansion for p is valid in this ease. The method therefore does then lead to a significant result*. Assuming the method applicable, and returning to the equa- tion * The disorimination between the cuses, and the explanation, are due to Hambni^er, Crdle, t. cm (1888), p. 342. y Google 284 we easily find SUBNORMAL 2ajao + Uj' — a, = y, and, for any value n which is greater than 2, 2(a„<i„ + «^.ai+...} + (jJ-3)a„_, = 0. If the constants in the equation were unconditioned, the co- efficients thus determined would give a diverging series for y. But we are assuming that the method is applicable, so that the conditions for convergence are to be satisfied ; and then, as .-(c+a ■■), where the last series converges. The method does not, however, give the tests for convergence of the series for y, at least without elaborate calculation : still less does it indicate that the con- vergence of the series for y is bound up with the polynomial character of the series in the expression for w. It can therefore be regarded only as a descriptive method, capable of partly indicating the form of integral when such an integral exists : manifestly, it is not so effective as Hamburger's. But the method, if thus limited in utility, has the advantage of indicating an entirely different kind of integrals of the original differential equation, which are in fact subnormal integrals, though it does not establish the existence of such integrals : for the latter purpose, other processes are necessary. It will be sufficient to consider an equation, say of the fourth order, in the form where the origin is a pole of p^ c Taking multiplicity ct^, for ^ = 1,2, 3, i. y Google 92.] INTEGRALS we have — = y +^yy +y^ — = y'" + ^yy" + ^y"' + ^'y + .'/'. so that the equation for 1/ is if + i^f + Sy" + Gfy +y' + p, (y" + 3i/y' + f) + Pi iy' + y^) + Pay + P4 = 0. If this equation is satisfied by aii expression of the form y = z--^ {a, + a,^ + ...), the coefficient of the lowest power of z must vanish. Now the governing exponents for the terms io succession are -TO- 3, -2m -2, -2m -2, - 3m - 1, - 4to, — wi — m — 2, — BTi — 2m — \, — cti — 3to, — OTs — m — I, — ^^ — 2m, To determine which groupings of terms will give the lowest power of z, we use a Puiseux diagram*; and in connection with each quantity ra-^ + km + 1, for the various values of fi, k, I, mark a point {ot^ + 1, k) referred to two rectangular axes Ote, Oy. Through the point (0, 4) take a line parallel to the axis Ox, and make it swing in a clockwise sense until it meets one or more of the points : round the last of the points then lying in its direction, make it continue to swing until it meets some other point or points ; and so on, until it passes through the point (in-,, 0). A broken line is thus obtained ; the inclination of any portion to the negative direction of the axis Oy being tan~^/i, the quantity ^n is a possible value of TO, and the terms giving rise to the lowest index of s in the differential equation for y are those which correspond to the points on that portion of the line. There are as many possible values of m thus suggested as there are portions of the line. y Google 286 EXAMPLES OF [92. It is not, however, a necessity of a Puiaeux diagram that only integer values of m. shall thus be provided : and it does, in fact, frequently happen that rational fractional values arise. Let such an one be - , where r and s are prime to each other ; and take When the independent variable is changed from s to %, an expres- sion for y <ii this type can be constructed, and it will be a formal solution of the equation ; if the series for y converges, then such an integral exists, expressed in the form of a series of fractional powers, and a corresponding integral w will be deducible. Such an integral, when it exists, is a subnormal integral. It is easy to verify that the only points, which need be marked in the diagram for the purpose of obtaining the possible values of m, are those which correspond with the quantities 4m, «r, + 3m, to-, + 2m, OTj + m, ra-,, as in § 86 ; but fractional values of m are now admissible in every case, instead of being so only under conditions as in the former use of the diagram, Ex. 1. This indication of integrals in a aeries of fractional powers was applied by Caylej and Hamburger, in the memoirs already cited, to the -(S-S)- which possesses neither a regular integral nor a normal integi'al in vicinity of 3=0. The only points to be marked for the Puiseus diagram are 0, 3 ; 3, there is one portion of line, and it gives Accordingly, we take and the oquation for w then becomea ^_ldw_ C^.^Cx or, wr.tmg w = f* W, " This equation ie used only for purposes of illustration; its integrals regular in the vicinity of s — <n , y Google 92,] SUBNORMAL INTEGRALS dHV ivhich is a special fonii of the earlier equatiou in § 91. It possesses two integrals, normal in (, if the quadratic (l(e + l) = 4y' + f hai> one of its roots an iiit«ger, that is, if y = TV(25-l)(25 + 3), where 6 is any positive integer {or aero). To tied the integrals, we have merely to adapt the solution in § 91, by taking a = 4p; ^ = 0, v = V + f = S(^ + l)- Thus ra = (.* = 3(3'*, o- = l, and ^(n-e){n + + 1)0^; and so, taking C5 = l, we have as a normal integral of the equation in f. Accordingly, the equation whore e is e I positi' ve integer or zero, and a ii 3 a constant, has . =.^i'"V i-K-iiJ ' {6+n)] .4- Manifestly, the other integral i s ^ven by „ ,^,-u.-s^ 'im 4, the two constituting a fundamental system, Eaeh of them is of the type of normal integral ; but the aeries proceed in fractional powers of the variable. It will bo noted that the two values of <r Bxe the same, and that only one yalue of p is used ; the relation is Sj:. a. Prove that the equation where X is a constant and 2fi is an odd int^er, pc two subnormal integrals. y Google EQUATIONS HAVING Equations op Higher Order having Normal or Subnormal Integrals. 93. There is manifestly no reason why Hamburger's method should he restricted to equations of the second order ; and he has applied it to obtain the corresponding class of equations of general order, the properties of the integrals defining the class being (i) the integrals are of the regular type in the domain of (ii) the origin is an essential singularity for each of the integrals, and at le^t ono of the integrals must be of the norma.l type in the vicinity of s = ; (iii) all the points, except z = and z = oo, are ordinary points of the integrals and the equation ; (iv) the number of zeros of at least one integral, which lie outside any small circle round the origin, is limited ; the second and the fourth of which are not entirely independent. Let the equation he of order n, and have its coefficients rational. The first of this set of properties requires the equation to be of the form where Pups, ...,pn are holomorphic functions of s for large values of z, and thus are expressible in series of powers of 3 of the form a^ + h^- + G^-^+ — (/i=l, ■■■,n). The third of the above set of properties requires that every value of z, except s = 0, shall be an ordinary point for each of the coefficients : and hy the second of the properties, s = is a singu- larity of the equation and therefore of some of the coefficients. Accordingly, the power-sfcries for the coefficients p, which have been taken to he rational and are limited so that every point except 2^ = is ordinary for them, are polynomials in zr^. y Google 93.] NORMAL OB SUBNORMAL INTEGRALS 289 As the integrals are regular in the vicinity of s = a> , one at least is of the form where Q is a series of powers of z~', which does not vanish when z=<x: and converges for all values of z outside an infinitesimal circle round the origin, and where p is a root of the equation p(p-l)...{p^n + -i) + a,pip-l)...(p-n + 2) + ... the indicial equation for 2=x. The exponents to which the integrals belong, being regular in the vicinity of z= 'x> , are the roots of this equation with their signs changed ; and they exist in groups or are isolated, according to the character of the roots. Let the above integral be one which, under the second of the set of properties, is a normal integral in the vicinity of s = 0, neces- sarily an essential singularity; in that vicinity, its expression is of the type where R {z) is a function of z, which is holomorphic in the vicinity of 2 = and does not vanish when s = 0, and where fl is a poly- nomial in «~^, say Ii = * - 1 3'«-i and (7 is a constant, Then, in the vicinity of ^ = 0, we have VJ z^+i s™ z^ s R (z) = r + B„ where T is a polynomial in - , constituted by U' -|- - , and i£i ie the holomorphic function of s given by R' {z} -r R (s). But as this arises through a form of the integral, postulated for the vicinity of z = 0, while the integral is actually known to be y Google the above form for v>'jvj must be deducible from this actual value. This is possible only if (^[-)i which has s = for an essential singularity, possesses at the utmost a limited number of aeros outside an infinitesimal circle round the origin ; for if it had an unlimited number of zeros in the plane, other than ^ = 0, any circle round the origin, however small, would include an infinite number, and then "<} would be incapable of such an expansion. The requirement, that thus arises, has been anticipated by the assignment of the fourth among the set of properties of the integrals; and so we may assume Q{-\ to have only a limited number of zeros. Accord- ingly, as in § 91, the form of 6 { - ) must be where -P(-) is a polynomial in - having as its roots all the zeros of Q (- j , and g[-\ is a holomorphic function of ~ , tinite every- where except at z=^. Let k be the number of zeros of Q ; then P f-j is a polynomial of degree k, an<l so it can be represented in the form where {z) is a polynomial in z of degree k. Thus the integral is of the form The postulated form must agree with this form; hence (I [-\ is the polynomial li of that form, and the holomorphic function R{p) of that form is the polynomial G{z) : also y Google 93.] INTEGRALS 291 The expression for w'/w in ascending powers of z is thus valid, under the conditions assigned, provided ii(s) is a polynomial in z. Taking so that Pi is a function of z, which is holomorphic in the vicinity of 3 = and is equal to a™ when 2 = 0, we have Then 10 \w) dz\w} = Z-^-^ {P,= + 2^Qx) = ^-''"T'P,, say. Similarly, "w = ^~"''" ^ ^'' '•' ^"' ^'* " ^"""""-Ps, say, and so on: where all the functions Pg, P3, ..., Qj, Qj, ... are holomorphic functions of z, and the first m terms in P, ai'ise from Pi". Substituting in the equation d'^w d'^^w dw ^2" -^ d3"~^ ' dz ' we have P„ + z'^ih Pn-l + 2'>=-P»-s + ■ . ■ + 2"™p« - 0, which must be identically satisfied. The coefficients p are poly- nomials in -: hence* z z-^p, is expressible as a polynomial in z, and so the highest negative power in p^ is z~'™ at the utmost. Accordingly, let J- ^ z z^ z'-^ Now we have F^ = rt,„-\- «™_,2 + ... + MiS""-' + ^2"= v + ^2", •^ If this were not the case, the aesignment of a larger value of m could Bet it: SDd GO the assumption reailj is no limitation beyond that wliicli is neoesBarj a noiTEial integral, viz. la must be a finite integer. 19—2 yGoosle 292 CONSTRUCTION OF [93. say, where T is a holomorphic function of z ; and where T^-x is a holomorphic function of s ; so that the first m terms in P,, which give all the coefficients in the exponent of the determining factor e", are given as the first in terms of a root of the equation u" + z™pi ii"~' + s™^5 !!"-= + . . . + Z^^'pn = 0, when the root is expanded in ascending powers of z. When the first «i terms in v are obtained, then the determining fa<;tor is known ; for we have li= {' ar"^-^v Moreover, after this determination, the terms involving the powers ^, z\ ..., ^"^' in have disappeared, so that this quantity is divisible by z™, leaving a holomorphic function of s as the quotient. 94. Having obtained the determining factor, let be substituted in the differential equation, which can now be taken in the form For this purpose, derivatives of e" are required. We have y Google 94.] NORMAL INTEGRALS 293 and so on, where v is identical with the first m terms of Pi, Vi is identical with the iirst m terms of P^, and generally, Vx is identical with the first m terms of Pk±^. Now ^ _ e" 2 I - J^ji^^"^ with the convention v„ = v, U-i = 1 ; and therefore the equation for u, after dropping the factor e^, is which can be written in the form where j)o= 1- The coefficient of m is Because the first to terms in ^a-i are the same as in P^, the first TO terras in the preceding coefficient are the same as in P™ + ^'"PiPn-i + . . . + ^""Pn, and they are known to vanish, for the coefficients of s", s'', ..., 2"'~' were made zero to determine v ; hence the preceding coefficient is divisible by a™, so that we can take if„_, + £"'piV„_^ + ... + z'"^p^ - 2^(0, + e,z + ...), where 6^ is a determinate constant, because v is known. The coefficient of 2™+^ -^ is dz = mUn-a + (Jl - 1) IJ,^3™^1 + . . . + Sv^f-" "i)^a + ^f"^^' "'p^i. The first m terms here are the same as the first to terms in mP„_i + {n- l)z"'p,P^, + ... + aPis'^-^'^^^H- af-^^iJ^-i, that is, the same as the first m terms in „„«-! +(n~l) v'^-^z'^p, + . . . + 2vz("-^^ "'p^-^ + z '"-" ^'pn-v yGoosle 294 CONDITIONS FOR EXISTENCE [94. The equation for v is H" + 3™Pj^"-' + z^"'p^v'"^ + ... + z"™p„ = ; and, in particufar, the equation determining a^, the constant term giving n values of o^, 95. Let a,„ denote a simple root of this equation, sometimes called the characteristic equation : then th^ quantity «£»«,''"' + (»~l)«™""'ni,» 4- ... + «„_i,«m-m is not zero. The coefficient therefore of z™^' y , as given above, does not vanish when s = r let it be '7o + '?i2+ -■-. where 5jo is a determinate constant, because v is known. It follows that the equation for it, in the form as obtained, is divisible throughout by s*^. Further, if it possesses (as, for the class of equations under consideration, it must possess) a regular integral, and if that regular integral belongs to the exponent a, then a is given by the' indieial equation V„<y + t^o = 0, so that u can now he regarded as a known constant. Further, we had where & is a positive integer (or zero), and p is a root of the equation p{p-l)...{p-n+X)^p(p-\)...{p-n^2)a,, + p{p-\)...{p-n-\- 3)aai + ... + /5a„_i,o+ano=0, say, of '(p) = 0. Consequently, the equation y Google 95,] OF NORMAL INTEGRALS 296 regarded as an equation in k, must have at least one root equal to a positive integer or zero : if this root be denoted by k, one condition that u should be of the form u = a" {c„ + c^e + . .. + c^if) (which is the fonn for u required by the earlier argument) is But while the condition is necessary, it is not sufficient for the purpose. When the value of u is substituted in the equation, the latter must be identically satisfied ; and so we have relations among the coefficients c. The general relation is /(o- + a) o.+g,(a)c^^,+g,(a)c^, + ... + £r™„^™(a) c,+™^™ = ; the relations for the first few coefficients are of a simpler form. When these relations are solved, so as to give successively the ratios of c,, Cj, ... to Co, a formal expression for u is obtained. In this forma! expression, all the coefficients c,+,, c^+2, ... are to vanish ; that this may be the case, we must (as in § 79) have /(ff + «)c, = 0, 7 (<r + « - 1) c,_i + 9, (« - 1) c, - 0, /(<7 + «-2)c,_ + ^,(«-2)c,_, + 5.(«-2)c, = a and so on, being m.(n — 1) relations in all. Of these, the first is known to be satisfied as above; it is the first condition for the existence of u in the specified form. The aggregate of conditions is sufficient, as well as necessary: the last of them secures that c,+i vanishes, the last but one secures that c^+j vanishes, and so on : the first secures that c»+mn-m vanishes ; and then, in virtue of the general difference-relation among the constants c, every succeeding coefficient vanishes. Thus when the m (ji — 1) conditions arc satisfied, in association with a simple root of the equation a normal integral of the original equation exists. It may happen that the conditions are satisfied for more than one of the simple roots of the equation : then there will be a corresponding number of normal integrals of the equation. y Google 296 NORMAL [95. The extreme case would be that in which every root of the equation is simple and the conditions are satisfied for each of the roots ; there then would be n normal integrals. Let the n roots be denoted hy 6^, 8^, ..., On, so that, if the normal integrals will be of the form where Uy is a polynomial, say of degree Xr, in s- We have (Tt = p -- Kt ; when these n indices a-,, arc associated with n quantities p, it follows that for r = l, ...,». The distinct quantities p^ are the roots of /(p) = 0, so that, if they are all different from one another, we have n of them ; also i_(„, + ,,) = i„(,>-l)-a,.. The value of 2 o-^ can then be obtained as follows. Construct the fundamental determinant \ dz ' dz ' ■"' dz which is equal to that is, to rft-. where A 1 1 constant. Now if w^ = 1 + c„j yGoosle 95.] INTE(3RA.LS 297 where u^, is a polynomial in z which is equal to 1 when s = ; also where Un is a polynomial in s which is equal to 1 when e = ; and so on. Thus „+^„+..-i.,.- )(.+i)(i.(^), 2). 1 +... 1 + e, +... 9, + «,■ + ... «,"+,.., ... As the roots are unequal to one another, <I>(z) does not vanish when 3 = 0; and it is a polynomial. We thus have -*"l''-'"'"+"3><ii)-^«~-( Accordingly iii + ... +fl„ = — + ... + 1 Ci.ir ^S<r,-i»(»~l)(m + l) = -«,, that is, "l" (s) reduces to its constant non-vanishing terra. Thus i^„,-i»(»-l)(m+l)-o,.. We saw that I (,r, + «,) = i»(n-l)-«,.; which is impossihJe because no one of the integei's k, is negative. It therefore follows that when the characteristic equation y Google 298 MULTIPLE ROOTS OF [95. has all its roots distinct from one another, and when the quantity/ denoted by u has n distinct values, associated respectively with n distinct roots of I(p) = 0, the differential equation where cannot have more than n — 1 normal integrals, linearly independent of one another. If, however, the quantity denoted by a liaa fewer than n distinct values, so that it could he the same for more than one of the n distinct quantities li, the relation l^^. = i«(«-l)(m+l)-a,„ would still hold, repetitions occurring on the left-hand side. But in that case not all the roots of the equation / (p) = are specified, for the same value of le could be associated with the value of a- common to two integrals; and the relation S(,T + «) = i«(»-l)-a.. no longer holds. The theorem then cannot be inferred as neces- sarily true : and it will appear from examples that an equation in such a case can have a number of normal integrals equal to its order. Similarly, if o- has n distinct values, and if these values are not associated with n distinct roots of / (p) = 0, the preceding theorem is not necessarily true; the differential equation can have a number of normal integrals equal to its order. 96. Next, let a„ denote a multiple root of the characteristic equation a™" + Om""' ffli, im + ■ . ■ + c[»,«™ = ; then the quantity ?jo vanishes, where lJ, = )iam"-^ + {>l — l)a,^"'^«i,im + ... +an-i,wm-m- The indieial equation is and cr must be a finite quantity. If 0„ is not zero, the latter y Google 96.] THE CHARACTERISTIC EQUATION 299 condition is not satisfied: and tlien the original equation lias no normal integral to be associated with that multiple root. If B^ is zero, the preceding indicial equation is evanescent : and so further consideration is required. The differential equation for «, on division by 3™+', becomes dn (»,+ 9,J+. + 2-+> (*. + *, wh. ere the coefficient of ',-^- is of the form for r = 3, 4 2'-+'"'~"(t. + +.^ + When ni^l, the indicia] eqnation is when m > l.the indicial equation is In either case, we can have a possible value for cr. A regular integral of the equation for ti, and a consequent normal integral of the original equation, exist if the appropriate conditions, corresponding to those for a simple root, are satisfied : it is manifest that they become complicated in their expression*. 97. It might happen that, in determining v, one or more roots of the equation «7«" + «!»""' «!, im + ■ ■ ■ ■+ (^i, nm — is zero, while some of the remaining coefficients in v do not vanish ; the implication is that (other conditions being satislied) a normal integral exists, having a determining factor of which the exponent is a polynomial with a number of terms less than m. It might even happen that, with a zero value of a^, all the associ- able values of the rest of the coefficients are zero, so that v = 0, and the determining factor disappears. One possibility is the existence of a regular integral, and the possibility can be settled in the particular case by the method given in Ch. vi. If, however, the conditions for a regular integral are not satisfied, then there is the ' They are considered by Giinther, Cnlle, t. cv (1889), pp. 1—34, in particular, pp. lOetseq. yGoosle SUBNORMAL [97. Hty of a subnormal integral of the original equation ; it IS follows. Lot «.=(«,. substituted in the equation d'Hv "dz^ ^ !£!-- fj»=o; en the equation for \ .,. is (by § 85) S+fi ^--)|S- ... + g.«. ■0, lere d ?,- f,. Now fi is to be chosen so as to diminish the multiplicity of a = as a pole of g„. After the preceding hypotheses, we shall not expect to have an expression of the form (-1/ _ Uj , ^ I , _^~''i__ where m is an integer; but after the indications in § 92, it is possible that II' may be a series of fractional powers. Accord- ingly, assume that the multiplicity of s = as an infinity of li' is /t, so that ^''fi' is finite when 3 = 0: then in ^n, we have a series of terms with infinities of oi-ders n/i , (n- l)/i-|- 1 , ... (ji-l)^+ m + l, («~2)^ + m + 2, ... (?i. -2)^+ 2m + 2, (n-3)y^ + m-|-3, ... n{m + \). Construct a Puiseux tableau by marking points, referred to two axes, and having coordinates 0, « ; \, n-\\ ... yGoosle 97.] INTEGRALS 301 (it is easily seen to be necessary to mark only the first in each row), and construct the broken line for the tableau, as in § 92. If the inclination to the negative direction of the axis of y of any portion of the line is tan"' 0, then ^ is a possible value for ft. If 6 be a positive integer ^ 2, we have a case which has already been dealt with. If ^ = 1, there may be a corresponding integral ; but it is regular, not normal. If ^ be a negative integer, li' is not infinite for 3 = 0, and the value is to be neglected. If ^ be a positive quantity but not an integer, it must be greater than unity to be effective ; for if it were less than unity, H would not be infinite for s = Q. Suppose, then, that 6 has a value greater than unity; as it arises out of the Puiseux diagram, it must be commen- surable : when in its lowest terms, let it be where q and p are integers prime to one another, and q>p. Then take 3 = a* ; we have an equation in u and x, and a possible determining factor e" can be found such that and so a series of fractional powers. The investigation of the integral of the new equation in u and x, that may exist in connection with this quantity fi, is of the same character as the earlier investi- gations. Equations of the Third Order with Normal or Subnormal Integrals. 98. The preceding general theory, and the methods of > with the cases when the equation for a^ has equal roots, or h zero roots, may be illustrated by the considei'ation of an equati< y Google 302 EQUATIONS OF [98. of the third order more clearly than by that of an cqua.tion of the second order, as in § 91. Taking the ainiplest value of m, which is unity, the equation is of the form ", ^^lo^ + ^ji '/ k«,sf + kill -^ k-,« , w +3 , — w H — — — -; w fC^Z -j- iC^S + jCj^Z + fi^aa r. , which, on using the substitution ti,„z+k„ ^^ iu y = we ^ =iuz''">e ', becomes where the constants a are simple combinationB of the constants k. The substitution adopted changes a normal integral of the one equation into a normal integral of the other, save for the very special case when it might be changed into a regular integral of the other: it therefore will be sufficient to discuss the form which is devoid of a term in y". In the present case, m = 1, we take y = e'u, and a is chosen so as to make the coefficient of the lowest power in the coefficient of w equal to zero. We thus have and the equation for u then is + -, [a^e'' + («si - a«™ - 6a) a + (a,, - ««,. - 6^0) = 0, of whicli the indicial equation for s :^ is It is clear that the equation in a. will not have a triple root : if it could, we should have (Xs = 0, «3a = 0, a = 0, the last of which y Google 98.] THE THIBD OBDER 303 values leads to the collapse of the process. (Account must, of course, be taken of the possibility that csja = = a,^, and this will be done later.) Meanwhile, we assume that a is either a simple root or a double root. First, let a be a simple root ; then o^a + 3a^ is not zero, and the foregoing indicial equation then gives a proper value for a-. If — p ia the exponent to which an integral in the vicinity of 0—x> belongs, p is a root of the equation /{p) = p(p-l)(p-2) + a„p + «,, = 0. The general investigation has shewn that this must have a root of the form p = o- + k, where k is a positive integer (or zero), and that, if this condition is satisfied, the form of u is M = 3''(C„ + C,S+ ... +0,3"). We substitute this value, and compare coefficients. If ff„ = {ff + n){ff + n-l)(<7 + ii-2) + a^(a- + n) + a^, k„ = -SaX<T+n)(a + n + l)+{a,, + 6o.){^ + n + l) + Ug — aoan — 6a, then the difference -equation for the coefficients c is for values of m^O, together with As a is a simple root of its equation, a^, + 3a^ is not zero : thus all the quantities k^,, k^, kj, ... are different from zero, and the pre- ceding equations thus determine Ci, C3, ... in succession, say in the form In order that the integral may not become illusory, the series is to be a terminating series : it would otherwise diverge, on account of the form of §„■ Let the series contain k + 1 terms ; then all the coefficients c»+i, c,+a, ... must vanish. Now c,+i vanishes if then c,+a vanishes if y Google 304 NORMAL INTEGRALS OF AN [98. and then all the succeeding coefficients c vanish. The latter condition gives ^,= which, as ^„=/((r+«,) for all values .>■' n, is the same as /(„ + «).0, a known condition ; and the other gives which is the new condition. When hoth conditions are satisfied, a noiTTial integral exists for the equation in y. As that equati i ■ involves seven constants, which are thus subject to two conditio:!., there are effectively live constants left arbitrary, subject solely :■■ - a condition of inequality as regards the roots of the equation moreover, « may be any positive integer (or zero). If the corresponding conditions hold for a second simple root of this cubic equation, the number of independent constants is reduced to three, while there are two integers such as k; the differential equation for y then has two normal integrals. If all the roots of the cubic equation are simple, and the corresponding conditions hold for each of them, there are three integers such as k, and there is effectively one arbitrary constant : the differential equation for 1/ would then have three noi'mal integrals. This, however, is impossible, if there aie three diff<!renfc values <r, a', <r" of a, and three associated integers k, k, k", yiich that (7 4- K, <t' + ic', <!■" 4- k" are different roots of f{p) = 0. For then Now we have loo? + attji - ^35 where dk da y Google 98.] EQUATION OF THE THIRD ORDER hence, summing for the three roots of h, we have by a well known theorem in the theory of equations. We then should have the equation which is impossible as no one of the integera k, k', k" can be negative. Hence, when the equation a? + tto^B — Osj = has three distinct roots, and when there are three different values a, a, a" of IT, associated with three integers k, k, k", such that <r + «, (r'+ k, o-"+ k" are different roots o{f(p) = 0, then the differential equation cannot have more than two normal integrals. But, if the values of <j- are fewer than three in number, or if the quantities a- + k are not different from one another, then the differential equation (the other conditions being satisfied) can have three normal integrals. Next, let a be a double root of the equation ft. = a* + aiXffl — asa = 0, BO that we have in order that this may be the case, the relation must be satisfied. The quantity a-, given by (3a^ + £%) (7- + o^y ~ aos, — 6a^ = 0, is infinite, unless a^ — aaj, — 6tt' vanishes : if this condition is not satisfied, then the regular integral for the w-equation, and conse- quently the associated normal integral for the y-equation, cannot exist. Hence a further condition for the existence of the normal integral is, that the equation Ksi — noai — 6a^ = be satisfied, where a is the double root ; that is, y Google 306 EQUATION OF THE Assuming this to be satisfied, the equation for u aow is Now Sctj, 21, say; so that the equation for u is The indicial equation for z — is Substituting « = 3«(p„+C,3+... +C„3''+.,.) in the equation, we have Du = Cc^S {C,,f (fl - 1) + C.,ye + Osij, provided for all values of )i ^ 0, where g^=^{e^n)(e^n^l){8 + n-2) + a^{e+n) + a^, A„ = c,„(S + m + ])(^ + m) + C^(^ + ?t + l) + c„. First, let the roots of the indicial equation be unequal, say X and /i, so that Du=c,o,„e^{e-\){e- (>.). Then the value of u, when 8 — X, gives an expression which formally satisfies the equation ; but it has no functional significance unless the series converges. That this may happen, g^ must vanish for some value of n, say k^, when d ~X; that is, one root of I(p) = p(p-l)(p~2) + a^p + a,, = must be pT=\+ K-i, y Google 98.] THIRD OllDER 307 where «i is a positive integer or zero. If that condition is satisfied, then a regular integral of the w-equation and an asso- ciated normal integral of the ^/-equation exist. Similarly, if l{p)~0 has another root p = fi + «:.„ where «, is a positive integer, then the value of u, when $ = /j., has significance. It is a regular integral of the w-equation; and a corresponding normal integral of the original equation then exists. Let denote the root of the cubic that is simple : then the earlier investigation shews that a corresponding normal integral may exist. If a' be the exponent to which the regular w-integrai belongs and H k^ + I he the number of terms it contains, then the equation /(p) = has a root p = a' + >c,. But the three normal integrals, each one of which is possible, cannot coexist, if X + ati, /j, + k^, ct' + k, are different roots of /(p) = 0, supposed not to h&ve equal roots. If they could, we should have >. + /i + (7 -j- Ki+ IC^ + Ks — ^p — S. Now X4-w = l- — = 1 - — . Also and for a, a, are the roots of the equation a^ + aa^i — agj = ; on reduction, after using the value of a and the relation fflisa — "«ai — 6a° = 0. Hence \ + fi + <r'^5, and therefore /C| -i- /Cs + «3 = ~ 2, y Google 308 NORMAL INTEGEAL8 OF AN [98. which is impossible, as no one of the integers « can be negative. Hence, when the roots of the indieial equation C„.r(,r-l) + C„.r + «. = are unequal, acd wheo /(p) = has not equal roots, the original equation cannot have more than two normal integrals, unless (in the preceding notation) there are equalities among the quantities X + Ki, /i+Ka, cr'+zcj. If it possesses the two normal integrals associated with X and fi, it is easy to see, from the expression for kn, that, if \ — /j. be a positive integer, it must be greater than Ka + 1 : and that, if /t — X. be a positive integer, it must be greater than «i + 1. Next, let each of the roots of the indieial equation for <r be equal to r : so that Thus the two quantities ."(^-t)^. M- [!],..• are expressions that formally satisfy the equation : they have no significance unless the series converge. That this may happen, g^ must vanish for some value of n, say k, when d — r; that is, one root of the equation I(p) = p{p- 1) (p - 2) + a,„p + «,„ = must be p — t + k, where «' is a positive integer or zero. (The quantity A„ never vanishes in this case and so imposes no condition.) On dropping the coefficient Co, the expression for u in general is equal to ■''/i„A/' ■■■^* ^^AA. so that the two integrals are of the form V, vlog^ + iJi, where v — [u]e=T, and Vi is an expression similar to v with different numerical coefficients, viz. the coefficient of (— lyz^'^'' in v, is [hX -.*,-, Ui \g. ~tie K dei]\,„ ■ The corresponding normal integrals are <fv, «■(» log « + ..). y Google 98.] EQUATION OF THE THIRD ORDER 309 A third normal integral can coexist with these two in the present case in the form where u belongs to the exponent c', = — + 4, provided l{p) = has a root of the form a-' + x^, where «, is a positive integer (or zero). The reason why three can coexist in this case is that only two quantities t and (/ arise, and only two roots, not three roots, of I (p) =0 are assigned. B^\ 1. Prove that, if the equation possesses a, normal integral of tlie form the constants j3, n, o- are given by the relations (T (3S= + oao) + 3a=S - 9|3^ 4- ("Is, + /3aj2 + ctaa = ; and the equation p{p-l)(p-3) + p«j4 + <%'=0 must have one root equal to tr+K, where ic is a positive integer (or zero). Obtain the relations sufficient to secure that the series Cg+OjB+.,. shall contain only k + 1 terms. Assuming that three values of a, distinct from one another, correspond to three seta of values of a and ft prove that their sum is 9 : and hence shew that, in this case, the differential equation cannot hive more than two normal In what circumstences can th i fl t 1 i n i tl 1 integrals ? En). 2. Obtain the constant and tl e nd t ns of e ten e f the normal integrals of the equation n the p e ed ng e a nple when a an she? and a^ does not vanish. How m ny n mal tgl ntlequtn then have? 99, We now have to c d ( ) tl e ca n 11 ne ze o root for a occurs, so that a^ — 0; and (ii), the case in which all the roots a are zero, so that a^ = 0,0.^^ — 0. Taking a^ = 0, the equation is / + zi — 2/ + —3 — — y=u. y Google 310 srEoiAL [99. Two non-zero roots arc given by a normal integral may exist in connection with each of them. The indicial equation for « = is in connection with this exponent, a regular integral may exist. The investigation of the respective conditions is similar to pre- ceding investigations. Now substitute in the equation the equation for u is u"'+3u"n' + u'Uil''' + il"-i a^± a^iZ + Oi^X _ _ and by proper choice of il, the multiplicity of a = as a pole of u is to be diminished. Assume that z~i'-Df is finite {but not zero) when z = 0, and form the tableau of points in a Puiseux diagram corresponding to 3|U, 2/4 + 1, iM + t, /1 4- 4, 5, that is, insert the points 0,3; 1,2; 2, 1 ; 4,1; 5,0. The broken line consists of two portions : one of them gives /* = 2, the other gives fi = \. The former gives the possibility of two normal integrals : the latter gives the possibility of one regular integral as above. But now let a23 = 0, as well as Us^ = 0. The equation for a becomes a= = 0, so that the method gives no normal integral. When we proceed to the equation for u, the coefficient of it is yGoosle 99.] CASES 311 We form the tabloau of points in a Piiiseux diagram corre- sponding to d/j., V + 1. f^+^' /' + -''. ^: that is, we insert the points 0,3; 1,2; 2, 1 ; 3,1; 5,0. There is only a single portion of line ; it gives /i = f. Accordingly, we change the independent variable by the relation s = a^; the form of il' is of- «* that is, dll_3a' W'_0 ^ rfa; ~ ** "•" a^ "a^'^x'' say. The differential equation with the substitution y^vx', becomes + ^ {t1a.^ + {tla,, + 180=,) ^ + {^-la^ + 18<i™ + 8) a^j = 0. If a determining factor exists, then (Ex. 1, § 98) it is of the form ^ + 2?o.. = 0, that is. /3- a. 3^= + 9(12,3 = 0, - (tjittg Substituting y Google [99. and using these values of a and /3, we find the equation for u i the form dx'- + " [- 9;8= + {o? + 63aa + 27a,0 x + |(12a,„ + 4) /3 - 6a^] a^^ + a (9a„ - 2) it^ + (27tt™ + 18a» + 8) a^] = 0. If the equation in ^ is to have a normal integral, this equation in u must have a regular integral belonging to an exponent cr, where it is easy to see that The regular integral for -u is of the form u = X c,,a:^+\ If /„ = {n + 3) (« + 2) (n + l) + (9a™ -S){n + 'S) + 21a,, + lSc(,„ + S, g^ = 3a(H + 4) {n + 3) - 6a (« + 4) + « (9<i» - 2), A„ = 3/3 (■« + 5) (« + 4) + (3a^ - 9,9) (n + 5) + {\2a.^ + 4) ^ - 6a^ & = a=+63a.s, + 27aai, i„ = 3^^(^ + 4), the difference-relation for the coefficients c is together with fe(, + LiCi = 0, A-aCfl + tei + Z-jC; = 0, ^_iC„ + A_iCi + ACa + Li^s = 0, The conditions, necessary and sufficient to ensure that the aeries for M terminates with (say) the (« + l)th term, which is the generally effective manner of securing the convergence of the series, k being some positive integer or zero, are four conditions in all. y Google 99.] CASES Assuming these satis6ed, we have M^/i 2 ( a subnormal integral. If the conditions are satisfied for more than one of the cube roots of djg, then there is more than one integral of subnormal type. Moreover, the value of ct is the same for all three cube roots, and only one value of « is required : so there may be even three sub- normal integrals, each containing the same number of fractional powers. In order that this analysis may lead to effective results, it is manifest that «3a should not vanish. Hx. 1, Prove that the equation possesses three subnormal integrals. Ex. 2. Discuss the integrals of the equation Normal Integrals of Equations with Rationai. Coefficients. 100. In the discussion at the beginning of this chapter, the only requirement exacted from the coefficients was as regards their character in the vicinity of the singularity considered: and a special limitation was imposed upon them, so as to constitute Hamburger's class of equations in §§ 91—99. More generally, we may take those equations in which the coefficients are rational functions of z, not so restricted that the equations shall be of Fuchsian type ; we then have Po jiiT -^Pi -JZazT + ■■■ +P»w = 0, yGoosle 314 roiNCARfi ON [100. where pa, P\, ■■■, J>n s^rs polynomials in s, of degrees to-,,, ot,, .... w,; respectively. The singularities of the equation are, of course, the roots of po = and possibly ^ = oo ; owing to the form of all the other coefficients, it is natural to consider* the integrals for lai^e values of \z\. It will be assumed that the integrals are not regular in the vicinity of z = oo . When a normal integral exists in that vicinity, it is of the form where is a uniform function of z~' that does not vanish when s = x , aJid O is a polynomial in z of degree (say) to, so that the integral can be regarded as of grade m. As in ^ 85—87, the value of ii' is obtained, by making the m highest powers in the expression i)oa'" + p,0'"-' + ... +p„ acquire vanishing coefficients; and a Puiseux diagram at once indicates whether a quantity fl' of such an order can be con- structed. The value of m — 1 is the greatest among the magnitudes provided two at least of them have that greatest value, which may be denoted by k. Then for such normal integrals as exist, we have when k is an integer, and where [h] is the integral part of h, when h is not an integer. The integrals are of grade ^h + 1, or ^ [h] + 1, in the respective cases ; and the equation is of rank k + 1. Take the simplest general case, when the equation is of rank unity, and when, in the vicinity of 3= co , it may possess n normal integrals which, accordingly, must be of grade unity. No one of the polynomials^, ...,pa is of degree higher than p„; assume the degree o?po to be k, and let * See Poincar^, Jnier. Joiirii. Math., t. vii (1885), pp. 30.^—268; Acta Math., t. vjij (188(5), pp. 295—344. y Google 100.] NORMAL INTEGRALS 315 where some (but not all) of the coofficients a may be zero and, in particular, where it will be assumed that a^ and «„ differ from zero. The determining factor for any normal integral is of the form e^' : 6 satisfies the equation fr,(f) = a„^"-l-a.(9''- + ...+fl„_.^ + a„ = 0. The preceding theory then shews that, if the roots of this equation are unequal and are denoted by 8^, 9^, .... 6^, the normal integrals are of the form the quantities cr^ are given by the equations where and ^i,(jh,---, ^nare uniform functions of r"', which do not vanish or become infinite when s = x . Special relations among coefficients are necessary in order to secui'e the conveigence of the infinite series <p ; unless these conditions are satisfied, the foregoing expressions only formally satisfy the differential equation and, as integrals, they are illusory. Ei^. 1. Prove that the equation possesses three normal int^rals in the vicinity of j; = co , when a is a positive integer not divisible by 3 ; and obtain them. .Kk. 2. Prove that the equation pOBseisses three subnormal int^rals in the vicinity of x= re , when -(-9' m being an integer not divisible by 3 ; and obtain them. (Halplien.) .Ec, 3, Skew that the equation has two normal integrals in the vicinity of 3^='xi ; and, by obtaining them, verify that the points a;= 1, x^ ~i are only apparent singularities. (HaJphen.) y Google EXAMPLES [100. Show that the equation one integral, which is a polynomial in x, and two other integrals, normal in the vicinity of a; = to . (Halphen.) Ex. 5. Prove that, if normal integrals exist for the equation the constant a must be the product of two consecutive integers. (Halphen,) Sx. 6. Prove that, if all the singularities for finite values of 2 which are by tbe integrals of the equation are poles, and ii pt,Pi, ..., ^„be polynomials ins such that the degree of p^ is not leas than the greatest among the degrees of pi, ,.., p„, then the primitive of the equation can be obtained in the form where the constants qj, ..., q„ are determinate, and all the functions i^i, ..., <^ are rational fiinctions of s. (Halphen.) Ex. 7. Apply the preceding theorem in Ex. 6 to obtain the primitive of the equation where n is an integer ; also the primitive of the equation (ii) „-+'-:?'rf-('-?'+o)„.o, where ii is an integer prime to 3. (Halphen.) fii'. S. Similarly obtain the primitive of the equation aT5(a^-l)y'-3^-e(^+.»:i'-l)y=0, in the form (MatK Trip., Part ii, 1895.) y Google 101.] Laplace's definite integral 317 PoiNGARfi's Development of Laplace's Definite-Integral Solution. 101. Several instances, both general and particular, have occurred in the preceding investigations in which formal solutions, expressed as power-series, have been obtained for linear differen- tial equations and have been rejected because the power-series diverged. These instances have occurred, either directly, in association with an original equation, or indirectly, in association with a subsidiary equation, when an attempt was made to obtain regular integrals of an equation, some at least of whose integrals were not regular; and they have arisen when an attempt has been made to obtain normal integrals of an equation, which is of the requisite form but the coefiicienta of which do not satisfy the latent appropriate conditions. In such instances, the expressions obtained for formal solutions do not possess functional significance. But Poincar^ has shewn that it is possible to assign a different kind of significance to such solutions in a number of cases. In particular, there is a theorem*, due to Laplace, according to which a solution of the given diifer- ential equation with rational coefficients can be obtained in the form of a definite integral; this solution has been associatedf by Poincare with the preceding results in § 100 relating to normal integrals. For this purpose, let where the contour of the integral (taken to be independent of e) will subsequently be settled, and T is a function of t the form of which is to be obtained. If this is to be a solution of our equation, we must have or, if r7, = 6,(»-)-6,t«-'-f... + 6„ * See my Treatise on Differential Equahong § 140. + In the memoirs ijuoltd in the fdolnote un p 314. The following exposition s based paitly upon tlieie memciri, paiOv npon Picard's Cti-ars d'Jnalyse, t. ill, y Google 318 Laplace's [101. the necessary condition is ({U^z" + Ujz"-' + ... + Uk) e'-'l'dt = 0. Let F,-."TO.^-.'-.|(TO._,)+...+(-l)'-|a('''^.-). forr=l, 2, ...,h. Then for each of the h values of r ; and the value of [e*^ F",] depends upon the contour of the definite integral. Using this result, the above condition becomes [le"r,]+/.»{TO-|(m-.) + ... + (-i)>|.(m)}.i< = o, which will be satisfied, if T bo a solution of the equation and if the contour of the integral be such that The equation for T is „ d'r /, ill, „\ di-'T so that its singularities are the roots of (/„ = 0, that is, are the points d,, Os, ■■; ^n, and possibly infinity. Writing the equation in the form the value of Pj when t is infinite is - on. Further, the quantity S Vy involves derivatives of T up to order k-l inclusive. This equation for T has its integrals rcgnlar in the vicinity of each of its singularities 6^, 6^, ..., $„: their actual form will he y Google lOJ.] DEFINITE INTEGRAL 319 considereil later. Let "^^ denote the most general integral of the equation for T in the vicinity of Sg; then, assuming that the conditions connected with the limits of the definite integral can be satisfied, we have an integral of the original differential equation in the form ^je''-^,dt, and this result is true for s= 1, 2, ..., n. Now '^g is certainly significant, because it is a linear combination of & regular integrals of the equation for T; hence we have a system of w integi'als of the original differential equation. 102. This system of k significant integrals can be transformed into the system of n normal integrals, when the latter exist. They can be associated with the formal expression of the n normal integrals, when the latter are illusory. A preliminary proposition, relating to the given differential equation, must first be established*. In the first place, let it be assumed that all the constants in the equation for T are real, and that T and t ai'e restricted to real values. That equation can be replaced by the system dt " When we substitute r,. = 0^e-*', (r=0, 1, ...,k-l), with the conventions that T,, — T and %„ = 0, the modified system ^^ = - P^Q - P^-^e, - ... - P,0j._. - (P, - X) @fc. ' It is due to Llapounoff (lfi92|i see Hoai-d, Cours d'Anabju,X. ill. p, 36: y Google 320 liapounoff's [102. Hence i^(@'^ + ««),= + ... + 0v,)-?^(®=+ ©/+.-.+ ©Wj+C^-POeVi + ©Hi + ©.e, + . .. + «H)s_@i_i - Pft 00*^, - ... - p,ei_,0fc_3 . Take a real quantity (o, smaller than the least real root of [/"„ = ; as ( ranges along the axis of real quantities between — co and t^, all the quantities Pj, Pa, ..., P^ remain finite. Hence, by taking a sufficiently large value of \, the quadratic fonn on the right- hand side can be made positive for that range of values of t ; and therefore, as t increases from — «) to i„, the quantity steadily increases in value. Consequently, when ( decreases from (j to — =o , the quantity steadily decreases in value. As („ is not a singularity of the equation, the values of 0, @i, .... 0j_i for any integral that exists at % are finite there ; their initial values are finite, and therefore each of the quantities [0], j@,|, ..., |0v_i| remains finite and decreases steadily, as ( decreases from 4 to — o^ . Hence the quantities all remain finite within that range, that is, no one of them can become infinite, for a value of X sufiiciently large* to make the quadratic form positive. Next, suppose that the constants are complex, so that T, Ti, ... can have complex values ; but let t still be real. Then we write 's Differential Caieulus, Srd ed., p. 408. In the so that it is auiKeient to take \ greater thiin the greatest positive value which malies the leit-hand side in the last inequality vauisli. y Google 102.] THEOREM 321 for all values of r, where 0^ and i|r^ are real ; the system of equations takes the form t=*™. t=t.« (.=0,1 ,,-M Oa 0i_2 + q-i ■^i.-.i — ... — ^10 + q,,-^ where 0o=0, '^„ = ylr, and Ps=Ps + iqs- We now have equations; on substituting they give the modified set - 9^ •J' - j)j,^ = \ 2 (4>r' + ^r') + (X - Pi) (<['%-i + ^t"*-l) + bilinear terms. As before, by choosing a sufficiently large value of X, the right- hand side can be made always positive. Then, by taking a value io smaller than the least real root of ?7o = 0, and by making t decrease from ft, to — m , so that all the quantities p and g are finite, it follows that, for such a variation of (, steadily decreases, and therefore that each of the magnitudes remains finite within the range from % "^o — oo . Hence each of the quantities yGoosle 322 liapounoff's [102. remains finite within tlie range of t from („ to — oo , for a value of X sufficiently large to make the quadratic form positive. Lastly, let the constants be complex, so that T, Ti,... can have complex values ; and now let t be complex in such a way that, in the variation from to towards — aD , where ( = ff, -j- re"*, a remains unaltered. The independent variable now is t, a real quantity, varying from to — qo ; and the preceding argument applies, A finite number \ can be found such that each of the quantities remains finite within the range of (. But hence a finite quantity X' can be chosen, so that each of the quantities remains finite within the range of ( from („ towards - co . In the first and the second cases, let ^ = X + (T, where rr is any real positive quantity that is not iniinitesimal ; and in the third case, let where <r is any real positive quantity that is not infinitesimal. Then, because in the respective cases tend to zero, as t becomes infinite in its assigned range, it follows that a quantity //. of finite modulus can be obtained, such that all become zero when t becomes infinite in its assigned range. This is true, a fortiori, when fi. is replaced by another quantity of the same argument and greater modulus. y Google 102.] THEOREM 323 lb also is true when any one (or any number) of the quantities T ghoiild happen to be multipUed by a polynomial- in t. For all that is necessary is to take a value /i+p, where p has the same argument as (i ; then e'"P, where P is a polynomial in t, is zero in the limit, when ( is infinite in its assigned range. Thus a quantity /i can be chosen so that where P, Pj, .... Pi_i are polynomials, all become zero when t becomes infinite in its assigned range from 4, which is not a singularity of the equation, to — co , 103. This result is now to be applied to the equation which determines T. Let t=dr be any one of the roots of Ug ~ 0, and consider a fundamental system of integrals in that vicinity. If [ JJ 1 : + (fc-l) = U77 -1. the indieial equation for Or is Suppose that p is not an integer. The integrals which belong to the exponents 0, 1 , .... k--2 are holoraorphic functions of t— 0^ in the vicinity of 0^ (Ex. 12, § 40) ; and the integer which belongs to p is of the form (t - e,y F (t - Or), where P is a holomorphic function of its argument. The contour of integration has yet to be settled. In con- nection with the value 0^, we draw a straight line from that point towards — co , either parallel to the axis of real quantities by preference, or not deviating far from that pai'allel, choosing the direction so that the line does not pass through, or infinitesimally near, any of the other roots of [Tj = ; and we draw a circle with 6r as centre, of such a radius that no one of those other roots lies within or upon the circumference. The path of t is made to be (i) in the line from — co towards 0^, as far as the circumference of y Google 324 DISCUSSION OF [103, the circle, (ii) then the complete circumference of the circle, described positively, (iii) then in the line from the circumference back towards — oo . So far as concerns the conditions imposed upon T by the relation at the limits, we have only to take the values at the two extremi- ties (= - oo . Now V^ is a linear ftinction oi T,T^, ..., Tt-, , the coefficients of these quantities in that linear function being poly- nomials in z and (; hence, taking z as equal to the quantity ^ of the preceding investigation, or as equal to any other quantity of the same argument as fj. and with a greater modulus, we have at each of the two infinities for ( ; and so the conditions at the limits are satisfied. In these circumstances, the complete primitive of the equation for T is T^A(t~e;yF{t-e,) + Qit-er), where Q is a holomorphic function of t — dy, involving m — 1 arbitrary constants linearly. The corresponding integral of the original equation then arises in the form Tdt, taken round the chosen contour. 104. We proceed to discuss this integral for large values of |s|. Let a be the radius of the circle in the contour, so that the series P and Q converge for values of t such that \t — 0r\<a. For simplicity of statement, we shall assume* that the duplicated rectilinear pai't of the contour passes parallel to the axis of real quantities from t^-dr — a to i = — x. From the nature of the integral T, we know that a finite positive quantity X exists, such that the value of * The alternative would be merely to take it value of a, Knii then jnalie t" vary from - a to -m . f"" y Google 104.] THE DEFINITE INTEGRAL 325 remaina finite, as t decreases from 8r — ct to — co . Let £ denote the maximum value within this range ; then for all the values of (, and then Let z have the same argument* as X, and have a modulus greater than \\\, that is, with the present hypothesis, let z be positive; then the part corresponding to the lower limit is zero, and we have r"'e"rdt < -^ e<'-» '"'-"• , J -a, z — x for values of z that have the same argument as X and have a modulus greater than |X| ; aud S is a finite quantity. Similarly, if, after t has described the circle, S' denote the maximum value of e^*2' for fl^ — a >(> — <», thou the second description of the linear part of the contour gives an integral, such that r "e'^Tdt < ^-^ e''-» '^-"' , for similar values of e ; and B' is a finite quantity. If, then, these two parts of the integral be denoted by /' and /"' respectively, we have where a is a positive quantity ; hence for any constant quantity q, however large, we have Limit (s^e-'^'-I') = 0, when s tends to an infinitely large positive value. Similarly, in the same circumstances, we have Limit (s^e-^^'F") = 0. * This form of atatemeiit is suited also for the variation of ( indicatecl in the preceding note. y Google 326 poincare's discussion [104. Now consider the integral round the circular part of the contour. As Q(t — 8r) 's a holoraorphic function over the whole of the circle, we have taken round the circle ; and therefore the portion of the integral I e'-'Tdt contributed by this part is /", where /" = J (( _ e,)^ e'^P (( - 6,) dt, on taking A — 1. The function P is holomorphic everywhere within and on the circumference, so that we may take P(t-Or) = c, + c,it-$r) + ... + c^(t-erT + R„, where \R,a\ can be made as small as we please by Bufficiently increasing m ; for if g be the radius of convergence of P{t~ 8,), so that g>a, and if M denote the greatest value of \P{t~8r)\ within or on the circumference of a circle of radius c, where |ii™|<i/'- for values of ( such that it-$,\^a<c- The value of the integral taken round the circu inference can be obtained as follows. Draw an infinitesimal circle with $,. as centre, and make a section in the plane from the circumference of this circle to that of the outer circle of radius a along the linear direction in which t decreases towards — co . The subject of inte- gration is holomorphic over the area of this slit ring : and there- fore the integral taken round the complete boundary is zero. Let ' T. F., % 23. y Google 104.] OF LAPLACE'S INTEGBAL 327 J' denote the value along the upper side of the slit, J" the value along the lower side, K the value round the small circle which is described negatively ; so that J' = r {i - OrY e"P (* - Br) dt, J" = /"'■"""e-s-i. (f _ OrY e'=P (( - dr) dt and, if the real part of p be greater than — 1, then* Hence, beginning at the point on the outer circumference which is on the lower edge of the slit, we have r + J' + K + J" = 0, that is. Let u denote the integral U= { '' it- OrY ^^i, and consider the value of it, for large values of z. Let ( - f , = - T = re" ; then Taking real positive values of s, write so that, as z is to have very large values, the upper limit for % with the new variable is effectively + oo ; thus yGoosle 328 Laplace's solution [104. Also, if V denote the integral then V ^ {- ly e^'- r T^ e-" R^dr Further, IJ*^ y'-e-yR.ndy \ < M (^J"^' ^/„ V^'^^y which, when the real part of p + 1 is positive, can be made less than any assigned finite quantity as m increases without limit, because a < c. Using these results, we have r = f' J 2 c„ (( ~ BrY + -H J (f - dry e" dt when TO is made as large as we please, and the real part of p is greater than — 1. Hence /" is a constant multiple of this quantity. 105, If now Wr i^enote the integral of the original equation, we have ■Wr=U'Tdt = /■ + /" + /'", so that For very large values of z, the first term on the right-hand side tends to the value nero ; so also does the second term. The third is a constant multiple of i {-\yz-'c^Vip + a + \). yGoosle 105.] AND NORMAL INTEGRALS 323 Hence, dropping the constant factor, we have Wr = e'^^s-'-' i (- l)-s-c. r (p + a + 1), for very large values of s. If the coefficients, of which c^ T(p+a+l) is the type, constitute a converging series, then this expression has a functional significance. If they constitute a diverging series, the result is illusory from the functional point of view. Now we have P + l = dU, and therefore the preceding integral, when it exists, is of the form When the aeries converges, this expression agrees with the foi'm in § 100, which is where <l>r is a holomorphic function of a"' for large valnes of 2. It thus appears that, when Laplace's solution of the equation, originally obtained as a definite integral, can be expressed ex- plicitly aa a function of z, which is valid for large values of ^, it becomes a normal integral of the equation. This normal integral has arisen through the consideration of the root S^ of the equation U^ = 0. When the corresponding con- ditions are satisfied for any other root of that equation, there is a normal integral associated with that root. Hence, when n normal integrals exist, they can be associated with the roots of the equation U^ = 0, which comprise ail the finite singularities of the equation in T. Note. It has been assumed that p is not an integer. When p is an integer, logarithms may enter into the expression of the primitive of the equation for T, and they must enter if p has any one of the values 0, 1, .... k — 2. There is a corresponding inves- tigation, which leads from the definite integral to the explicit expression as a normal integral. When the normal integral exists, y Google 330 EXAMPLES [105. it can always be obtained by the process in § 100. If logarithms enter into the expression of e^u, they enter into the expression of u in the usual mode of constructing the regular integrals of the equation satisfied by !(. Es:. 1. The preceding method of obtaining the normal integral gives a test as to the convergence of the series in its espr^sion. If the infinite converges, which must be the case if the expression for the developed definite integral is not to prove illusory, its radius of convergence r is given by the relation* Lim|c„r(p + a + l)r = ^. But, from Stirling's theorem for the approximation to the value of r In), when n is infinitely large, we have %+c^{t-6;,+c,,{t-e;f-\-... must conven^e over the whole of the i-plane ; and therefore the integral TV is of the form wtere ^ (() is holomorphic over the whole plane ; a I'esult due to Poincard Ex. 2. Prove that, if the condition in Ex. 1 ia satisfied, a normal int^ral certainly exists. (Poineard.) Ex. 3. Consider Bessel's equation for large values of |i;|. The int^rala in the vicmity of j; = oo may be normal — they are not regular — and, if normal, must bo of gradi" unity. Accordingly, let then the equation for a is We take y Google 105.] bessel's equation 3 and theu seek for a regular integral (if any) of the equation j^m" + {^ + 20^) u' + (to -v?)u = 0. If an integral, regular in the vicinity of a: = =o, can exist, it is of the form Substituting, and making the coefficients in the resulting equation vanish, we have c^{26p + 6) = 0, anii, for all values of m, «=j + So„, + i(2p-2m-l)-0. and the latter then gives Hence, taking 60 = 1, and 6 = i, a formal solutioa of the original equation is and taking S= — i, C|,=l, another formal solution is If 2ji is an odd integer, positive or negative, both series terminate ; and the forma] solutions constitute two normal integrals of the equation. It is not difficult to obtain an espression given by Lomaiel* for ■/„, in a form that is the equivalent of IfSn. is not an odd int^er, both aeries divei^o ; and the formal solutions are then illusory as functional solutions. When Laplace's method of solution is adopted, so as to give an integral of the form the equation for T is On writing (i^ + l) T" + ZtT' + {l -^ n^) T=0. V independent variable, the equation for T is y Google BB8SELS EQUATION AND [105. This is the differential equatioi elements are given by e function, wliose (Gau o + fi + l=3 a^=l Thp contoui cf the integral oonsi^tR of (i) a <,irde ruund i as tPiitre with radius leas tlnti 2 ('o as to exclude — ;, the otlier finite singuHnty of the equation m T), and then (ii) a duphcited hue fium a point in the Liruum feience pissing m the direction of a diameter tontinued towards co The ai^ument of / and the argument of j: muit he -^uth that the real pajt of xt is negative. In order to construct the integral, we need thf complete primi- tive of the ^-equation in the vicinity of ii = 0: it is where A and B are arbitrary constants. The part multiplying A, being a holomorphic function, merely contributes a zero term to w ; and we need therefore substitute only the other part. Manifestly, we may write B=l, Now ^(a-y + 1, j3-y+l, 2-y, ^) = i^(<i-i, (3-^, h ") 1 2m + l , „ 2m + n(m+^)-n(-^); Taking this value of c^,, we substitute in the definite integral. In the preceding notation, we have d.=i, p=-i, <r,= -(p + l)=-i n(m-<r,) = n(m + 4); so that, when the solution where t=i-2iv, is expanded into explicit form, ifc becomes a constant multiple of y Google 105.] DEFINITE INTEGRALS 333 But ^n(,H)^ "«'-"'»'-:l-'"-*''-'' n(-i). SO that, after substituting for c« and rejecting the constant factor n(-^), the integral becomes a constant multiple of m=o ni! {2iJ^)"" which agrees formally with the expression earlier obtained. The corresponding integral, associated with the primitive of the ^-equation in the vicinity of i= — i as a singularity, can be similarly deduced*. E^: 4. Shew that the equation where «,'^ - ia^ is not zero, can be transformed to xy/' + {\, + \ + 2)'y/ + {x+i (\i^\ti)}w = 0. Assuming Xi, X^i ^i + ^a "^ot to be integers, prove that the latter equation is satisfied by for an appropriate contour independent of x ; and deduce the normal series which formally satisfy the equation. (Horn.) Double-loop Integrals. 106. Before proceeding further with the investigation in §§ 101 — 105, which is concerned partly with the precise determ- ination of a definite integral satisfying the linear differential equation, we shall interrupt the argument, in order to mention another application of deiinite integrals to the solution of certain classes of linear equations. It is due to Jordanf and to Poch- hammerj, who appear to have devised it independently of one ' In connection with the aolation of Bessel's equsition by mes^ne of definite integtals, papers by Hunkel, Math. Ann., t. i (1869], pp. i67 — 501; Weber, ib., t. 3X3V1I (1890), pp. 404—416; Macdonald, Proc. Lond. Math. Soc, t. sm (1898), pp. 110—116, ib., t. XXI (1899), pp. 165—179; and the treatise by Graf u. Gubler, Einleitung in die Theorie der Bessehchen FunUionen, (Bern), t. i (1898), t. n (1900) ; may be consnlted. t CoiiT-B d'Aiwlyu, 2= k&., t. Ill (1896), pp. 240—276; it had appeared in the earlier edition of this work. + Math. Ann., t. xsxv (1890), pp. 470—494, 495- 536; II., t. sxxvii (18S0), pp. 500—511. y Google 334 nouBLE-Loop [106. another. A brief sketch is all that will be given here : fur details and for applications, reference may be made to the sources just quoted, and to a memoir by Hobson*, who gives an extensive application of the method to harmonic analysis. As indicated by Jordan, the method is most directly useful in connection with an equation of the form where Q (s) and zR (s) are polynomials, one of degree n, the other of degree ^niiiz, R {z) also being a polynomial. For simplicity, we shall assume Q (s) to be of degree n. Consider an integral W= { T {t - zY+''-' dt, where 2" is a function of t alone ; this function of ( has to be determined, as well as the path of integration. We have AW-(-l)"(a + 7i-l)(a + n- 2). ..(« + ]) + {t-zY\R{z) + {t-z)R'{z)^^^''R"{z)+JATdt = /[« (* - ^y-' Q (i) +(t-zrii im Tdt, the summation being possible because Q and R are polynomials of the specified degrees. The integral will be capable of simplifi- cation, if the integrand is a perfect differential ; accordingly, we choose T so that which gives TE(t)=^[TQ{t)\, Q(t) * Phil. Tmne., 1896 (A), pp. 443—531. y Google 106.] INTEGRALS 335 The preceding integral then becomes jdV, where mt) Hence the original differential equation will be satisfied if jdV^O; and this will be the case, if the path of integration is either (i) a closed contour such that the initial and the final values of V are the same : or (ii) a line, not a closed contour, such that V vanishes at each extremity*. Each such distinct path of integration gives an integral. It ia proved by Jordan that there is a path of the first kind, for each root of Q ; and that, when there is a multiple root of Q, paths of the second kind are to be used. Again, restricting Q (s) for the sake of simplicity, we assume that each of its n zeros is simple; let them be (ti, esg, ..., a^- As the polynomial R (z) is of degree less than n, we have M(fl = S - where 7,, ..., 7,1 are constants; and then To obtain the paths desired, take any initial point in the plane ; from it, draw loops^f round the points a,, ..., On, z, and denote these by A^, A^, ..., An, Z. Take any determination of * A third possibility would arise, if the patli were same value at its extremities; tut thia case is of yery n ■t T. F.. % 90. y Google DOUBLE-LOOP INTEGRALS [106. which is the subject of integration in W, as an initial value ; and let the values of W, for the various loops J.,, ..., j4„, Z with this as the initial value, be denoted by W {a^, ..., W{a^, W{z) respectively. An integral of the original differentia] equation will be ob- tained, if the path of integration gives to F a final value the same as its initial value. Such a path can be made up of ArAgAr~'^Ag-^, that is, first the loop Af, then the loop Ait, then the loop Ar reversed, then the loop jig reversed. Let W{ar, Wj) denote the value of the integral for this path ; then W{aT, a^ is a solution of the differentia! equation. Taking the above initial vahie (say /(,) for /, we have W{ar, a,) = W(ar) + e"^->r Wia,) - e="^*. W(ar) - W(a,) = {1~ e^'v,} W(ar) - {1 - e-^r] W (a,) ; for after the description of A^, the initial value of 7 is e^'^t./^ for the description of Ag-, it is e''"'>'r+''s'7t, for the description of Ar~', and it is e""^*./,, for the description of j1,~\ It is clear that [l-e'^yi] W (a„ ar) ^ [1 - e"^-'.} W (a„ at) + {1 ~ e'"^,] W{at,ar); and therefore al! these values of the integrals, for the various appropriate paths, can be expressed linearly in terms of any n of the quantities W(«r, Cs)> in particular, in terms of W(z,a,). W(z,a,), ..., W(z,a^). Each such quantity is an integral of the original equation ; and we therefore have n integrals of that equation. iVofe. For the special cases when a or any of the conetantB y is an int^er ; for the cases when § (I) has multiple roots ; and for the oases when B(t) ia of degree n — \, while Q{1) is of degree less than m — 1 ; reference may be made to the authorities previously cited. As already stated, all that is given here is merely a brief indication of the method of double-loop integrals. y Google 106.] EXAMPLES 337 Es:. 1. Consider the equation of the qu.irter-pei'iod in elliptic functions, Here we have 9W«(--l), -/'- t''{t-rf=r-'{t-\)'^ ■(i-iy^(t-z)--'dl, and tho path of integration has to be settled. We have W(lj==2J'dW, I dW, where a mai'ks the initial point of the loops. Hence H'(0, l) = 2W{0)-2W{l) = ij' dW, W[0, 0-2tf(0)-2F{2) = 4prfir; and thu.s two integrals of the equation are given by l^dW, tdW. The comparison with the known results is immediate. Ea;. 2. Integrate in tho same way the equation ,, „, rf% ^ dw , ' ' cb= dz whore a and 6 are constants. (This is another form of the equation (l-,.)2'-2(»+l).f+(.-«)(»+« + l),.-0, by Hobson {I.e.) for unrestricted values of the constants m and re.) V. 22 y Google 338 ASYMPTOTIC! [106. Ex. ;i. Prove that when the equation wheie /( i» I tuuittnt, 15 iiiiijectpd to thf tr wistorm xtion the tunstonned equation (nhich is of Fuchsnn ty]ie, § 54) can, under a LCitain condition, be trcat<yl bj the foregoing method ind issuming the condition to be ^atiified ohtaiii the mtegial Ej 4 ^pply the method to the equation apply it alao to the equation of the hypergeometric series. (Jordan ; Pochhammor.) Ex. 5, Apply the method to solve the equation {\-i')ie"-2zw'-lw = 0, for real values of s such that — 1 <3<1. Sliew that the equation is ti-aus- formed into itself by tiie relations (^-l)(ir-I) = 4, w{z + \f^W{Z+lf- and deduce the solution for real values of z such that 1 < z < gd . (Math. Trip., Part ir, 1900.) PoiNCABit's Asymptotic Representations of an Integral. 107. After this digression, we resume the consideration of the investigations in §§ 101—105. In those ca.ies wheu the infinite series in a normal integral diverges, the normal integral has been rejected as illusory from the functional point of view. There are, however, cases belonging to a general class which, vfhile certainly illusory as functions of the variable, are still of considerable use in another aspect : they are asymptotic representations of the integral, to use Poincar^'s phrase*. A diverging series of the form y Google 107.] KEPRESENTATIONS 339 is said to represent a function J(x) asymptotically when, if 8n denote the snm of tho first ra + 1 terms, the quantity „•{!(,) -S.\ tends towards zero when x increases indefinitely ; so that, when a; is sufficiently large, we have fl;"{i/"(a:) — S„) < e, where |e| is a small quantity. The error, committed in taking S„ as the value of J, is less than Lch smaller than that is, the error in taking Sn as the value is much smaller than in taking S„_i. (The definition, though stated only for large values of x, applies also to the vicinity of any point in the finite part of the plane, tnatatis mutandis.) The asymptotic representation is, however, not effective for all values of the argument of the independent vaiiable. If a;" {-/(«) - Sn) tended uniformly to zero for all infinitely large values of x. the function J(x) would be holomorphic, and the series would converge: the permissible values of the argument of the independent variable are therefore restricted. It is manifest from the nature of the ease that, when such a series is an asymptotic representation of a function, the series can be used for the numerical calculation of the approximate value of the function for large values of a: with a permissible argument: the error at any stage is much less than the magnitude of the term last included. Without entering upon any discussion of the question why a diverging series, which is functionally invalid, can yet, when it is an asymptotic representation of a function, be of utility for the numerical calculation of the function, it is proper to mention one conspicuous example of the use of such series, as found in their application to dynamical astronomy*. The normal series, derived from the solution of the equation as represented accurately by the definite integrals, are proved by Poincar^ to give this type of asymptotic representation of the * In particular, see Poinoare, HUcaniqiie Cileste, t. ii. y Google 340 NORMAL INTEGRALS AND [107. solution. For, denoting the solution by w, and tlie sum of the first m -t- 1 terms of the series by Siji, we have Now c where and iHl<l. Then, as before, we have r M u 7 0™+' J _li-Pr| c which is a multiple of by a quantity independent of a. When we take so that, as a is to have large values, the limits of y effectively ai'e to + CO , the last definite integral is a multiple of -I T — •T^y'-*''^"^e-ydy. This definite integral is finite. Denoting its value by 7, we have where a is a quantity independent of e, and 7 is finite. Hence, when 2 is sufficiently large, we have z™ (we~=*'-s''+' - Sm) < e, y Google 107.] ASYMPTOTIC EXPANSIONS 341 where | e | is a small quiintity ; and so we can say that S^ asym- ptotically represents we~'^'- £'•+'■, or we can say that the normal series is an asymptotic representation of the actual integral, the repre- sentation being valid (on the hypotheses adopted earlier) foi' large positive real values of ^. Note. For further diseuasioii of these asymptotic expansions in connection with linear differential equations, reference may be made to Poincar^'a memoir*, which initiated the idea. Among other memoirs, in which the subject is developed and new applica- tion.9 are maile, special mention should be made of those^f by Kneser, and thosell by Horn. Picard's chapterj on the subject may also be consulted with advantage: and a corresponding dis- cussion on integration by defiuite integrals is given by Jordan§. Ex, 1. Shew that the complete primitive of the differential equation in the vicinity of ;c = co , can be asymptotically represented by ('^ + "' + ,^^+-)«>«'^+(^o+t + 5 + -)^'"'*'' and hq, (9o ive arbitrary constants. (Kneaer.) Ex. 2. In the differential equation i(-*S)+(«+<')»-°- ifc^ is an arbitrary parameter, A, B, C are real functions of x and (with their derivatives) are holomorphic when ai^x^h; moreover, A and B are positive. Prove that a,ii int^ral of the equation, determined by initial values that are independent of h, is a holomorphic traoseeiidental function of k ; and shew that, for large values of k, its asymptotic expansion if of the form ,!,.(*,+*>+.^.)co.i,+(*l + J' + .,.).mi», whore ^a, (f",, <^j, .,., mi are functions of x, (Horn.) * Acta Math., t. vni (1886|, pp. 295—344, + Crelle, t. oxvi (1896), pp. 178—213; ib., t. txvii (1897), pp 72— lOS; lb., t CM (1899), pp. 267—275 ; Math. Ann., t. xlix (1897), pp. 383—399. II Math. Ann., t. xm (1897), pp. 432—473, 473—496 ; i6., t. l (1898). pp. 525— 556-, ib., t. LI (1899), pp. 346—368; ib., t. LH (1899), pp. 371—392, 340-362. X Coura d'Analyse, i. iii, ch. xiv. % Goiirs d'Analyse, t. iii, ch, n, § iv. y Google 342 RANK [107. Ex. 3. Shew thiit the equation has a solution of tlie form. where A^, Bj„ ace rational functions of k, and that it has an aaymptotic solution of the form (*.+|= + ...)o™fa+(^ + *J + ...).mfa, and indicate the relation of the Molutions to uue anofclier, (Poinearri : Horn.) Equations of Rank greater than Unity replaced by Equations of Rank Unity. 108. When the differential equation possesses, in the vicinity of 3 = O) , normal integrals which are of grade m, then, denoting the degree of the polynomial p^ by ■m^, it follows (aa in § 85) that the degree -m,. of the polynomial p, is such that the sign of equality holding for some at least of the < Also, if e" be the determining factor of any such integral, then Of is the aggregate of the first m terms in the expansion, in descending powers of z, of a root of the equation The existence of the normal integral then depends upon the possesion of regular integrals by the linear equation in u, where In the case where in = 1, the method of Laplace certainly gives the integrals of the differential equation, even wheu the normal series diverge ; but it is not applicable, when m is greater than unity. Poincare, however, devised a method by which the given equation is associated with an equation of grade unity: Laplace's method is applicable to the new equation, so that its primitive is y Google 108.] OF EQUATIONS 343 known : and from tliis primitive, an integral of the original equation can be obtained by means of one quadrature. The new equation is of order ra'"; and the investigation leads to an expression for w dz ' which, whefl it oxiata, can be obtained more directly by Cayley's process (§ 92), Poinear^'s method is as follows. Let the given equation be supposed to possess n normal integrals of grade m, say, in the form e"'<''.^(4 B"'^'>M^) e""'«0«(^); let these be denoted by /i (a), /^(j), ...,/„ (a). Let a denote a primitive mth root of unity, say e™ ; and consider, in connection with any integral /(z) of the original equation, a product !,="nV(«'^). Then y satisfies an equation of order ii™. which possesses ?^™ normal integrals /.(')/. («)/.(»'^) .■•/.(«--H where a, b, c, ..., k are the numbers 1, 2, ..., n or some of them, any number of repetitions being permitted ; and these normal integrals are of grade m. Lot and let the equation for y be where, if Q^ be of degree in z, then the degree of Qjc_r in general is equal to d + r{m—'l), because of the grade of the normal integrals. Owing to the source of the quantity y, which clearly is not changed if 2 be replaced by sa*, s being any integer, it follows that the equation for y must remain substantially unchanged, when this change of variable is made ; hence fc--r(s«')g-'-'^-'"" _ where \ is independent of r. y Google 344 BANK CH. .>'GED [108. Now let the variable be changed from z to x, where then, because for all values ef «, the coefficients c„. being numerical, the equation for 2/ takes the form ■where The degree of -fi^y-j in ^, as it is determined by the highest terms in Q,v-a. i*5 which is independent of 3; so that the degree* of all the coefficients R is the same. Further, wc have B,_, (.«■) - i^ c„„ («')'->'-.'-'>--'e„_(««') for the power of a is thus Hence the equation is substantially unaltered, when z is replaced by 20* in the coefficients Jt ; hence, multiplying by a power of s, say z", where « 4 f + if (m - 1) = (mod m), i£ becomes a uniform function of x, when we substitute " Some mij-ht haye vanishing ooelflcieiits in particular cases; the argument deals with the genei'al ease. y Google 108,] HY TRANSFORMATION 345 The new equation is therefore an equation in the independent variable x such that all its coefficients are uniform,- They ail are of the same degree, so that it is of rank unity ; it has normal integrals, and some of its integrals may he subnormal. Laplace's method can be applied to this equation ; and we then have a solution in the form of a definite integral. The way in which this definite integral is used, in order to bring us nearer a solution of the original equation, is as follows. Let '.=/(^a"). (s = 0, 1, . -1), This has to be differentiated N(=n™) times, derivatives of w„, Wj, ..., w,„_i of order n being replaced, whenever they occur, by their values in terms of derivatives of lower order, as given by the diffei'ential equations which they satisfy; and, from the iV"+ 1 equations involving y, -j^ , ,.., -j-^, the iV" products de^ '"rfs* ■ d^ where a, b, ..., k « 1 can have the values 0, 1, . ,. , ., ..., k each can have the values 0, eliminated. The result is the equation for y. involving y, ^, ..., -^E^ can be regarded products of the type The N equations i giving these N "^~d^ each in I such be i of derivatives of y and the variables. Let two Assuming p known, as an integral of its own equation, the value of lUo is derivable by a quadrature. If y, first obtained as a y Google 346 POINCAR^'S METHOD [108. definite integral, can be evaluated into a functionally valid normal integral, it is of the form The function $ is linear in y and the derivatives of y, so that, when we substitute the value of y, we have ivhere "I* is^ free from exponentials ; and then 1 dw^ _-^ which can be expressed as a series in terms of z. The exponent to which it belongs is easily seen to be an integer, owing to the form of ■!> ; thus 1 <^M;n ™ , ^, „ a,„ «„,+i j-=(l(,3"'-' + ((l2™-^+ ... +(tm-i+ — H ->+-■■■ Wa dz Z Z^ But if y cannot be evaluated into a functionally valid normal integral, there may be insuperable difficulty in dealing with the quantity — . In instances, where the actual expression of a normal integral {if it exists) is desired, the process is manifestly cumbrous: as it does not lead to explicit tests for the existence of normal integrals, the simpler plan is to adopt the process indicated in |§ 85 — 88, which gives either a normal integral or an asymptotic expression for an integral in the form of a normal series. For further consideration of Poincare's method, reference may be made to his memoir, already quoted, and to a memoir by Horn*, who discusses in some detail the case, when the linear equation is of the second order and of rank p. Ex. 1. In the case of an equation of the second oi-der which is of rank 2, saj shew that, if w = 0(«), and if w^i^^{^-x), which will satisfy the equation d^w, , , , dw, i-«-^><-'>7Er+''><-"''"'-»' - Asia Math., t. xxiii (1900), pp. 171—201. y Google lOS,] EXAMPLES then ;i vai'iablo y., whore uniquely in terms <iiy. Tf, liowever, the invariants of the two equations are equal, so that of a quadratic equation, the coefficients of which are expre^ible in terras of y. (Horn.) TJx. 2, Discuss the equation for lai^e values of a-. (Poincare.) Ex. 3. Shew that, in the vicinity of r— o^ , the equation ormal integral of the second grade, when a is an odd positive Es^. 4. Ohtain the normal integrals of the equatioi^s (i) :k^" = (^ + |)2/, (ii) ^y = 2j;(l + &^)y + (;^^6V-26a;-j:)y, in the vicinity of a^=o; , y Google CHAPTER VIII. Infinite Determinants, and their Application to the Solution of Linear Equations. 109. In the investigations of the present chapter, infinite determinants occur. These are not discussed, as a I'ule. in books on determinants ; a brief exposition of their properties will there- fore be given here, but only to the extent required for the purposes of this chapter. Their first occurrence in connection with linear differentia! equations is in a memoir* by G. W. Hill : the convergence of Hill's determinant was first established f by Poincare. Later, von Koch shewedj that the characteristic method in Hill's work is applicable to linear differential equations generally; with this aim, he expounded the principal properties of infinite detenninants§. The following account is based upon von Koch's memoirs just quoted, and upon a memoir]| by Caazaniga, Let a douhly-infioite aggregate of quantities be denoted by where i, k acquire all integer values between — co and + x ; the quantities may be real or complex, and they may be uniform functions of a real or a complex variable. They are set in an * First published ill 1977; republished 4c (a Math., t. viii (1886), pp. 1—36, t BvU. de la Sw. Math, de France, t. xiv (1886). pp. 77—90. I Acta Stath.. t. sv (1891), pp. 53—63 ; ib., t. nvl (1892—3), pp. 217—235. § Foi tocther cliscuBBion of theii properties and their applications to linear differential eiiuations see a memoii by tbe same writer, Aeta Mat},., t. xstv (1901), pp 89—122 II Anwih di Uai mat a *5er 2' t jcxvi (1897), pp. 143-218. Other memoirs by CazzaniKa lealin^ with the la.iai' subject, are to be found in that journal, &ei ' t i|18JS| pp s-i-14 S^ > t. II (1899), pp. 329-238. y Google 109.] INFINITE DETERMINANTS 349 array, so that all the quantities with their first suffix the same occur in a line, the values of k increasing from left to right, and all the quantities with their second suffix the same occur in a column, the values of i increasing from top to hottoni. We then have an infinite determinant, which may be represented in the form Constrnct the determinant An.n. where then if, as m and n increase indefinitely and without limit, i),n,» tends to a unique definite value D, we regard the infinite determ- inant as converging to the value D. In all other cases, the infinite determinant diverges. To secure this convergence to a unique definite value D, it is sufficient that, when any arbitrary small quantity S has been assigned, positive integers M and N can be found, such that !-0«.+p,n+s-0™,«|<s, for all values of m greater than M. for all values of n greater than iV^, and for all positive integers p and q. The aggregate of all the quantities for which i = k, that is, of the quantities ..,, a_i,_i, (X„,o, Ch.i, as they occur in their place in the determinant, is called the principal diagonal, sometimes briefly the diagonal ; and a constituent of reference in the diagonal, naturally chosen in the first instance to be aj.u. is called the origiv. Let then the infinite determinant converges, if the doubly- infinite converges, all values of i and k between — <» and + oo occurring ii the summation. To prove this, let p„..= n 1+ s |4,.,i y Google 350 COUVEllGENCE OF [109. and consider Let P,„,B be expanded ; by omitting suitable terms and chauging the signs of others, we obtain Dm,». Hence, taking I'm,.!, making al! the terms positive, and adding certain other positive terms, we obtain .?„,„. Similarly^ we can pass from D^^p^n+q to Pm+p,n+q- Now take i>m+p,n+9 — -Om,!! ; make all the terms positive, and add certain other positive terms, and we have |i).+,.„^,-A.,«|<|-P™..,«-H,--^..,«l- But, because of the convergence of the series the product P^.n converges when m and n increase without limit ; hence, assuming any arbitrary positive quantity S, however small, integers M and iV" can be determined such that Pm-tp,n+q - P'm,n < ^i for all values of to greater than M, for all values of n greater than N, and for all positive integers p and q. Consequently, for the same integers, we have and therefore the infinite determinant converges. Such a determinant is said* to be of the normal form. AU the determinants with which we have to deal are of this type. Next, the origin may be changed in the diagonal without affecting the value of the determinant. All the conditions for the convergence of the determinant with the new origin are satisfied; let its value be D', and let D be the value with the old origin. Then taking any small positive quantity 5, we can determine integers M and N such that |-t'--Om,«|<S, |i>'-i>V,„,|<^, ' von Koch, Acta Math., t. xvr, p, 221. y Google 109.] INFINITE DETERMINANTS 351 for all values of m greater than M and all values of n greater than N, the determinant D'^^n, being the same as il^^n, ao that, if agg be the new origin, m, = m- $, Wi = n + ft Manifestly, D^^^ can be chosen so as to include the new origin. Hence \D-n'\ = jD - »™,„- (D' - Z>V,,„)| <\D-D,„„\ + 'D'-})\„.,\ so that, in the limit when 8 is made infinitesimal, d = b: Similarly, the value of the determinant changos its sign when two lines are interchanged, and also when two columns are inter- changed: so that, if two lines be the same, or if two columns be the same, the determinant vanishes. Further, if the determinant be changed, so that the lines (in their proper order) become columns and the columns (in their proper order) become lines, the principal diagonal being unchanged, the value of the determ- inant remains unaltered. If, in any line in a determinant of normal form, each of the constituents be multiplied by any quantity ft, the value of the determinant is multiplied by /j. ; likewise for any column, and for any number of lines and columns, provided that the product of all the factors (when unlimited in number) converges. Further, if all the constituents in any line of a converging normal determinant be replaced by a set of quantities of modulus not greater than any assigned finite quantity, the new determinant converges. In the determinant D, let the line ao^j (the constitu- ents occurring for values of k) be changed, so that a^^j is replaced by *'t, where kll < A, A being finite ; and let D', !>'„_ a for the new determinant correspond to D, D„^„. For comparison with Z*'„,„ construct a product P,„,,(, where p,.,,..'n'|i + l|^,.,i|, 1 having all values from —n to +m, except i = 0. Then, when i>',„_,i is expanded, there occurs in P^.n Jt term corresponding to y Google 352 PKOPBKTrES OF CONVERGING [109. every term in D'm^n. tbe latter having gome one factor x^ that does not occur in ^m.n\ hence I term in i?;„,„|^^ jterm in P™,„|. Now some of the terms in D'™,„ are negative, while all the terms in Pm,-n. are positive; and terms arise in Pm,n. the terms corre- sponding to which do not occur in D'm,n- Hence Similarly, where h can he chosen as small as we please, becanse i |i + I \AiA is a converging proiluct. The resnlt, which is due to Poincare, is thus established. Properties of Oonvergisg Infinite Determinants. 110. The development of an infinite determinant can be deduced from the preceding properties. We have -n I "tji, -11+1 • ■■■! 'hn.t = 2 ± (»_„__.„«._„ ^.,,_„+l . . . am.ni) say. In this expanded form, let ai_i = l + Aij, ai^k'^-^w (' + *); and let every term in the new expression be changed, so as to have a positive sign and so that each factor is replaced by its modulus. The resulting expression is greater than |Sm,„|; and every term that occurs in it is contained in P,„,«, where _ m ( m ) yGoosle 110.] INFINITE DETEEMINANTS 353 Also, Pm,« contains other terms, all of which are positive ; thus |S„,,|<-P,..,. Similarly, _ _ for all positive integers p and q. But P,„,„, with indefinite increase of m and m, is a converging product; hence 2m,n. in the same limiting circumstances, converges absolutely. Thus the usual method of development of a finite determinant holds in the case of an infinite converging determinant of the normal form, and «-[»»] l]-Z} = 2...a_2,j,,tt_,,p,ao,y,aj,g,as,g^ ... (_lY..+ (Pi-2) + (J'i-" + (J>o-"> + Wi-il + i5)-"l+- the sura being extended over all the permutations • ■■. p.., Pi, p«, ?i. 9^, ■■■ of the integers .... -2, -1, 0, 1, 2, .,.. Writing for all values of i and k, we at once have the expansion Z) = l + 2Ji,( + 2|^(,i, Aij\ + t\Ai_i, Ai,j, ^i,i ] + ..., ^Aj^i, Aj^jl -4j,i , J-jj, A}^ie\ the summations being for all integer values from — gc to + co such that i<j<k<.... 111. It follows from the preceding expansion of a converging determinant D of normal form that, when a constituent o^j enters into any term of the expanded form, no other constituent from the line i or from the column k enters into that term. Taking the aggregate of terms {each with its proper sign) into which Oi^ic enters, theii- sum may be denoted by ra; ^a^j ; and the determinant may be represented in the form y Google 354 MINORS [TU. or in the form The quantity Wj^i is called the minor of an, and sometimes it i denoted by It can be derived from D by suppressing the line i and the column k, or, what is the equivalent in value, by replacing Oj^j by 1, and every other constituent in the line i or in the column k or in both by 0, and then multiplying by (— 1)'"*. Manifestly, we have It is an immediate corollary that 0= S aj^tai.k, I (»4=i)[: k=-'B I for the right-hand side in the iirst is equivalent to i> with the line * replaced by the line j, so that the latter is duplicated ; and in the second, the right-hand side is equivalent to D with the linej' replaced by the line i, so that the latter is duplicated. More generally, if, in the lines and in the columns ft. A ft-, we replace all the terms by 0, except «a„B,, «o„p,, --., aa,.,e,., each of which we replace by 1, and then multiply by the result is the coefficient of y Google 111.] OF FINITE ORDKB 355 in jD. It nianifcstly is a minor of order r; and it is denoted by Clearly all the minors of any finite order are determinants of normal form, converging absolutely. If D is not zero, some at least of tlie minors of constituents in any line must be different from zero, and some of the minors of constituents in any column also must be different from aero. Similar results, when D ia not zero, hold for the minors of any order r of finite determinants, which are constructed out of r selected lines and any r columns, oi' out of r selected columns and any r lines. Further, the minor r + l. ..., 0, 1, r+1 0, 1, tends to the value unity, as r and s increase. To prove this, let Q,,~n{l + X\A„]]. where the product is for all the values of p, and the summation is for all the values of q, that are excluded from the ranges p = — r to + s, q — ~rto-^s. Expanding the minor, and changing every term so that its sign is positive and each fsictor in the term is replaced by its modulus, we have a new expression every term of which is contained in the expanded form of Qa,>-', and Qg^,. contains other terms. Further, the expanded minor contains the term +1 as does Q,^,., and all other terms involve the quantities A ; hence |(::; :::5::;;; 3 -i|< «..-'. But the product uh + iiA^A converges ; and therefore, when any small positive quantity S is 1, integers — r and s can be determined such that Qs.r - 1 < S. yGoosle 356 EXl'AJiSION Of [111. Taking these as the integers defining the minor, we have 80 that i-s<|(:::::;:;::::;:):<i+s- Moreover, as integers s', r are chosen, greater than s and r and gradually increasing, the quantity decreases ; and thns the minor tends to the value unity as r and s increase. One or two properties of minors may be noted. We have \k, l) \k, l) \l, k) \l,k)' for the changes from one of these expressions to another are equivalent to an interchange of two lines or an interchange of two columns, each of which changes the sign of the determinant. Similarly for minors of any order. Again, expanding ai^^ by reference to constituents of a column, we have and expanding it by reference to constituents of a line, we have Similarly, !, because it is me; also when q is neither k nor I, because it is a minor of the first order with two columns the same ; also y Google 111.] INFINITE DETERMINANTS whon h is neither * nor j, because it is a i with two Hnes the same ; and 357 .■ of the first order \k, I) = 0. where h is neither i nor j, and q is neither k nor I, because it is a minor of the first order with two columns the same and two hnes the same. Similarly for minors of higher order. The similarity in properties between finite determinants and converging infinite determinants of normal form is not exhausted by the preceding set : in particular, infinite determinants can be multiplied, and determinants framed from minors of an infinite determinant are connected with their complementary in the original, exactly as for finite determinants. The simpler of these properties are contained in the following examples. Ex. I. If are oonvei^ing doterminants of normal type, itnd if for all values of i and t, then c-[»...l s a convoi'ging determinant of normal type, and AB = C. Ex. 2. If ai,i, denote the minor of o^t in the determinant %k.' ■■■' %k. "i-k,' *■■' %i; I with the preceding notation for miDora of order ;■. Ex. 3. In connection with the determinant y Google S5S INFENITE DETERMINANTS AS [111. prove that Qfe|)+©(t»)-^©(*;S-(U;l)''^ QO;:S+©(U)*(')(l;i)-C:*;:t)"' and, more generally, that L f%\fH J H 1 .... ^ \_(hii h, h, ■■■! ir\ A .iAU\h-.i, K*ii, -, -f-n-i/ W. *i- ^, ■-. V where, in the typical term, k^, it„ + i, f« + 2, ..., ifcn-i preserve the aame cyclical order as fc^, ifcj, ij, ..., i^- In the first of these, the right-hand side vanishes if i is equal to ^, or l;^ i in the second, it vanishes if i is equal to t, or 4 ; in the third, it vanishes if k„ is equal to any one of the quantities tj, ^3, ..., i,.; and so in other cases. 112. The infinite determinants which arise in the discussion of linear differential equations have, as their constituents, functions of a parameter p. The preceding results are still valid, if the condition that is an absolutely converging series is satislned; in particular, the determinant converges absolutely, and its value may be denoted by D (p). The parameter may be made to vary ; and then it is important that the convergence of D{p) should be not merely absolute, but also uniform, in order that it may be differentiated. Suppose that, in any region in the p-plane, all the functions Aij(p) are regulai' functions of p, such that the series converges uniformly and absolutely. For all values of p within that region, any small quantity B can be assigned, and then integers M and N exist, such that for all integers m^M, and integers —n^ — N, 11 '1 A.cj{p)\<&. By analysis that follows the earlier analysis practically step by step, wo then infer that, for all integers m'^M, n'^N, and for all positive integers p and q, and for all values of p within the region indicated, we have -D,„,,.+,<p)--D„,.(p)|<28; y Google 112.] FUNCTIONS OF A PARAMETER 359 SO that l>(p) converges uniformly. Hence, within the domain considered, I) (p) is a regular analytic function of p. The expansions of D (p) in terms of its constituents have been proved to converge absolutely, by comparison with the expansions of Pm,w. where converges uniformly and absolutely, Pm.n is a product that con- verges uniformly with indefinite increase of m and n. The corresponding modifications in the investigation lead to the conclusion, that the expanded I'orra of TJ(p) converges uniformly as well as absolutely. Moreover*, this expanded form can be differentiated, and its derivatives are the derivatives of D (p). In particular, we have dp 9ffl,-,i dp = Z2, a. . -^^ . ■ dp Thus if D vanish for a value p' of p, and if all the first minors of D vanish for that value, we have ^1 = 0, while -;r^ is not iniinite; the first derivative of the uniform dp function D vanishes, and therefore p' is at least a double zero of B. In that case, we have d^D -^ d^ai I- .p.-^- 9a; 1 3"; t dai , j-i-So... dp dp -s^ss(';{ 9af,t dojj Hence, if all the second minors of D vanish for that valui we have dp- ' The proof is aimilur to those given for preceding propositions ; see to. y Google 360 INFINITE SYSTEMS [112. and so p' is at least a triple zero of D. And generally, if all minors of all orders up to r — 1 inclusive vanish, but not all minors of order r, when p — p, then p is a root of D in multiplicity r ; and B is then said to be of characteristic r. The quantity r cannot increase indefinitely, for we have seen that minors of sufficiently high order tend to the value unity, so that the general vanishing of all minors of the same order is possible only for finite orders. But it need hardly be pointed out that the converses of these results are not necessarily true: thus p = p' might be a double root of D, while not all the first minors of D would vanish. 113. The purpose, for which infinite determinants are to be used in this place, is in connection with the solution of an un- limited number of equations, linear in an unlimited number of constants. Let and suppose that the infinite determinant B, where converges uniformly ; it is required to find the ratios of the quantities x to one auother which satisfy the equations Ui = 0; (i = - ^ to + 00 ), the quantities a; being themselves finite, so that we have where X is finite. We know that fG)- converges absolutely; its value is B when j = k, and is when / is different from k. Moreover, the series S<\s is an absolutely converging series, and hence for values of o! considered, we have y Google 113.] OF EQUATIONS 361 where TJ is finite. Hence, by one of the propositions already establishei^, the quantity S, defined by the equation also converges absolutely, so that for all the other terms give a zero coefficient for x. Hence, if Mi = for all values of i, and if we are to have values of ic^ different from zero, then D = Q, which is a necessary condition. We shall assume this condition to be satisfied. If some at least of the first minors are different from zero, then the equation shews that any one of the quantities u, which it contains, is then linearly expressible in terms of the others, and so the correspond- ing equation w = is not an independent equation. Let m, then be omitted on this ground ; we have ?(i,'iV=ffG; ")"'■'''■'■ where on each aide the summation is for all values of i except 1 = 0. The coeificient of x^ on the right-hand side is ?(i,' ")"•■'■ This is zero, if q is different from both k and I; it is = I ; and it i; =fu: Jc. V ?((,'*)«"-"•■'■ y Google 362 if g = k. Thus INFINITE SYSTEMS [113. U i) = -«ll,t^i + «(,(■»* ■ But all the quantities v^ vanish ; hence We thus have for all values of /; and ^ is any finite quantity, for only the ratios of the quantities x are determiuabe. Similarly, if D be of characteristic r, so that the minors of lowest order which do not all vanish are of order r, let he snch a minor different from zero. We then have Thus the coefficient oi x When q is equal to any one of the integers ^-,, ^i, ..., 0r, this coefficient is equal to a minor of order r — 1 and so vanishes. When q is not equal to any one of those integers, the coefficient is equal to a determinant with two columns the same, and it is therefore evanescent. Hence ,S = 0, and therefore :::;:)" -T A. .... /3„, A, /3„ ■:::.> where, on the right-hand side, m must not be equal to any one of the integers a„ ...,ar. It thus appears that there are r relations among the quantities m; and that, in particular, each of the quantities m.^, u,^, ..., ■«„,. is linearly expressible in terms of the y Google 113.] OK EQUATIONS 363 remamiiig cjiiantitiea u. Accordingly, we assume these r quanti- ties u omitted from consideration. Denoting by a any integer other than a, a,, and by any integer other than ^j, ..., ^r, we have =(*:*::.■:;«:)'"'-&",&;:;;:»:) '''-(A.sT.r.'.A)'''-' in the same way as for the simpler ease ; hence, as all the quanti- ties !t„ vanish, we have U.A J"'-.!, (a. A A._ SO that all the quantities xg a,rc linearly expressible in terms of r such quantities. For further properties of infinite determinants, reference may be made to the memoirs quoted at the beginning of | 109. Ai'j'LicATioN TO Differential Equations. 114. When the differentiai equation is given in the form the substitution ~l /'^•''^ W = we ■' leads to an equation of order n in w, which is devoid of the term involving -r— ^ • The coefficients of the new equation are linearly expressible in terms of Q^, Q,, ..., Qn-i, Qn, and the expressions involve derivatives of Qj up to order n — 1 inclusive and integral powers of Q,. We may therefore take the differential equation in the form ^w-i.^ ' (/s"-= "■ " ' de y Google 364 INFINITE DETERMINANTS APPLIED TO [1.14. We assunie that, in the vicinity of 3 = 0, it possesses no synectic integral, no regular integral, no normal integral, and no subnormal integral. The point ^ = is then a singularity of the coefficients ; and, if it be only an accidental singularity {of order higher than s for Pg, in the case of some value or values of s), the conditions for the existence of a normal integral or a subnormal integral are not satisfied. We assume the coefficients P still to be uniform functions of s, and we shall suppose that their singularities are isolated points. Let an annulus, given by Ji<\z\< R, be such that its area is free from singularities, no assumption being made as to the behaviour of the coefficients P within the circle of radius R; then it is known* that each such coefficient can be expanded in a Laurent series P.= ic,,^^^ (r = 2, 3, ..., n), which converges uniformly and unconditionally within the annulus. Without loss of generality, it may be assumed that Ji<l<li': for, otherwise, we should take a new variable Z = e(RR')^, and the limiting radii R and R' of the annulus for Z then satisfy the conditions R<l<R'. Further, owing to the character of the convergence of P^, we have dP, - d ( dPr\ ? and so on; all these series converge uniformly and unconditionally within the annulus. Hence also y Google J 14.] LINEAR DIFFERENTIAL EQUATIONS 365 similarly converges within the annulus, where R (fi) is any poly- nomial in /J. ; and therefore, taking the circle I e\ = ]., every point of which lies within the annulus, the series _i|iiWc,,„j converges, 115. From the general investigations in Chapter ii, it follows that the equation certainly possesses an integral of the form J/=ZP<P (3), where p is any one of the values of g— . logo), the qnantity w being a root of the fundamental equation associated with an irreducible (but otherwise simple) closed circuit in the annulus ; and the quantity is a uniform function of z. As the integral is not regular, the number of negative powers of 3 in is unlimited ; and so we may write In order to have an adequate expression of the integral, the quantity p must be obtained ; the value of a„ h- a,,, for m = + 1, ± 2, ..., + X , must be constructed; and the resulting series must converge for values of s within the annulus. We first consider the formal construction of the expression for the integral. Let <i>(p) = p(p-l)...(p-n-t-l) + c,,_,p{p-l)...(p-n + S) + Cs,-3p(p-l)...(p-n + 4) + ... + C„^,,-n+iP + Cn,-,,; +(p + /i)...(p+>(-m + 4)c,,r_^_5+ ... ...+(p+/J.) C„-.,r-«-„+, + Cn.r-^-n ; and write Gm(p)=0(p + m)a™+rC^,^a^, where, in the last summation, the values of fi are from — x to + CO, with /j. = m excepted. Then we have p(,) = i G,(rt ..*--, y Google INTEGRAL TN THE FOEM OF [115. SO that y is an integral of the differential equation if for all values of m from — oo to + <» , there being no assumption that the negative infinity is the same numerically as the positive infinity. Let n for all values of fi other thai ■^in.m with the convention ») ti; and introduce a quantity ffm (p) = (m + p) S -^ni. where the summation now is for all values of y:*. We then requi the infinite determinant "((>)-[*.,,] ■ - ■ T ''i''— 1 — y ) ^ » ^—1 * ^— I 1 ' '^—1 ' ..., ^1,-2 1 ''/'"l,— 3 ■ ''/'"!, II I 1 ' '"/''l,! the necessary and sufficient condition of the convergence of which is the convergence of the double series for all values of m and fi between — co and + oo except m = f*. 116. In order to establish the convergence, we firat transform the expression of Cm.,.- Let then we may take (p+t^)(p+^l-l)...(p+^l-p + '\) = (p + m-X){p+m-\-l)...ip + m- = (p + my + «p,. (P + "O"- + V. ip + '«)' \-p + \) yGoosle 116.J A LAURENT where a,,,, is a polynomial in X of degree r. Using this for all the terms in G^ „, we have where A,(X) -+A,{\)(p + «.).- + ... + i,.W, Accordingly, w e have ^2S <^ (p + m) 1 W,(x)j + SS !*((> + »•) 1 !^.W! + Now the series 'i" |fl{i)o,. + S2t : byk" (Ml- converges for every value 2, S, ..„ n of r, where ! ii(\) is any polynomial in X. Hence every term of which (for the various values of p) converges, because ««-ii,s-j!+s is a polynomial in X of degree s — p + 2, and therefore the whole of the right-hand side is a converging series. Accord- ingly, we may write 'F A,{X) = H„ (s = 2, ...,n), and then each of the quantities |ffs| is finite. We thus have '.(p + mTjl ,|(p + m)^ i!?t..,.|<i^.i!rii^h™!ri^i + ... "' . I * (p + m) Assuming p to be any quantity, different from any of the roots of any of the equations *(p + m)-0, y Google 368 INFINITE DETERMINANT [116. each of which is of degree n, we know that all the series converge absolutely, for the values k—2,S, .... n. Moreover, the sum of each such series is a function of p : and then, if p varies in a region no point of which is at an infinitesimal distance from any of the roots of ^(p + m), the convergence of the series is uniform. Accordingly, the double series converges uniformly and unconditionally ; and therefore the infinite determinant ii(p) converges uniformly and unconditionally, pro- vided p does not approach infinitesimally neai- any root of any of the equations 0(p + m) = O. Clearly, il(p) is a uniform function of p, for such values of p. Further, we ha.ve ^«.,.(p)0(p+m) = C,„,„(p), and therefore t^+,,^, ip)<f>(p + m + l) = t?^+,.«+, (p) = -.|r,„,^(p + l)^(p + l+m), so that Construct the infinite determinant 0.(p + I), and then replace each constituent -^m.^-ip + 1) by ^,„+,^„+,(p); the result ia to give the modification of il (p), which arises by moving eaoh column one place to the right and by depressing each row one place, in other words, by taking ^}r,^,(p) in the diagonal as the origin instead of ifpo „(/)). But such a change makes no difference in a determinant which converges absolutely ; we therefore have n(p + i) = n(p), or the infinite determinant fl is a periodic function of p. Lastly, by making p infinitely large in such a manner, that it does not approach infinitesimally near any of the roots of any of the equations .^(p + m) = 0. yGoosle 116.] MODIFIED 369 (which roots for different values of m differ only by real integers, 80 that if we take p=u + iv, where u and v are real, it will be sufficient to take v large), we reduce to zero every constituent that lies off the diagonal of il(p). As every constituent in the diagonal is unity, and every constituent off the diagonal is zero, it follows (from the law of expansion of an absolutely converging determinant) that Lim n (p) = 1, provided p tends to its infinite value in the manner indicated. Modification of the Infinite Determinant ii (p). 117. It is convenient also to consider another infinite determ- inant associated with fl (p). The equation G^ ip) = was taken in the form <l>(m + p)Xir^.^a^ = 0: and the infinite determinant ii(p) was composed of the constituents ^m.f If ^"^ infinite determinant were composed of constituents Ip {m + p) yJTm,^, then the row determined by the integer would have a common factor 0(m4-p); and thus there would be an infinitude of factors, the product of which either should converge or should be made to converge. Let pi, p^, .,., p„ be the roots of (j) (p) = 0, so that <f> (p) = (p - pi) (p - p„) . , . (p - p„), and therefore ^. To change this into a form suitable for an infinite convergin product, we multiply by with the convention S.(rt-1. As A^(p) remains finite and is not zero for finite values of p, v may replace the equation (?m(p) = by y Google 370 MODIFICATION OF THE [IIT. Now let for all valuPB of ?« except m = 0, and Xo.'^ n (p-p„); also let Then the equations between the constants a have the form In association with these equatio determinant consider the infinite (p)= X- J X-^-3 X->-i x-». X-. . X-l!. •■■ X-1.-2 X-J.-1 X'V X-. . X-.,.. ••■ X.,-. X«,-i X",o X... , x... , ... »,-. X>.-i »,. Xi.i . Xi.= . ■■■ Xt- %2,-l X.,. X=,i . Xs,B . ■■■ Taking the diagonal to be ..., X-^.-s, X-i.~" X'>.'" X'." %',!' ■■■• ™ require to establish, (i), the convergence of the series summed for all values of m and /*, except m = fi, from - ;» to + k and (ii), the convergence of the series S(x-,.-l), in order to know that the infinite detenninant D (p) converges. We consider first the double series %'Zxm,u- ^s* i,(p) = .n-*„(f).n. (m + 0). y Google 117.] INFINITE DETEHMINANT 371 The quantities p^, p-i, ■■-, pn are finite; hence, so long as p remains within a finite region that does not lie at infinity, there is & finite quantity K which is larger than any value of j/'m(p)| for values of p within that region. Hence, as .\<K'- when m is not zero. When m is zero, we have »,,=*.WC.,,-o..,. Proceeding exactly as with tlie series SS^m,^. in § 116, summing for all values of m other than zero, and for all values of fi other than m = /i, we have lf + " -+|fflX lp + » ■\tl.\t- every term of which is finite, and therefore is finite. Also l\(!..A^\HJ\p'-\ + \H.\\p'-'\ + which is finite, so that converges. Hence the double series Slimmed for all values of m and /t between — co and + c , converges. Moreover, all the series, which superior limits in the inequalities, converge uniformly withir region of p considered ; hence the double series converges formly. The establishment of the convergence of the series i(fc,.-i) except in the y Google 372 CONVEEGENCE OF [117. is simple. We know, by Weierstrass's theorem*, that tho series converges uniformly and unconditionally; so that, if »„,,. = x.,--i. the infinite product n(i + (?™,™) converges uniformly and unconditionally; and therefore^ n_(i + |9.,,|) converges. But n(l + |^™,,„|)<l+2|^,„,,„|; hence S |^m,inl converges uniformly, that is, the series _S(x,.,«-l) converges uniformly and unconditionally. The eoiivergonce can also be established as follows. Let =(-^'). and choose a finite positive integer p, such that, for values of p under c sideration, we have \p-p'\<P, where p' is any one of quantities pi, p-2, ..., fn- The sum of the terms J(x«.-I) is finite, and may be omitted without affecting the convergence: and we e sider the sum of the remaining tenns, for which we have \m]>p. We have y Google 117.] THE DETERMINANT and therefore 2>ii^|7'^|<| <- |.j / 1 p - f-' + -- -Pff^ \p-i'.r\^ »? " ^ 1 \9-pA- Now for all the values of m under consideration and therefore J |p-P"[ -^1 ip-P'L, - p+l p+l' ^.K^P^ -P,\\ so that we maj 'take '■-S-"- -?^)\ where 1f.|<i. Hence Xm. „=nM^ ttj 2,,{,-,,l" Now 2 (iir(p—p^)2 ia finite for all the values of puuder conaidenttion, and it is finite for all values of m if ji^ involves m; let ^denote the gieateat value of its modulus. Again, for any quantity 0, we have othat r writing A' jV! 1 iV=i ] <-i4'(4)"4:(4.)^ shewing that the si converges. y Google 374 EVALUATION OF THE [117. The infinite determinant I)(p) thus converges uniformly and unconditionally for all values of p in the finite part of its plane. Its relation to li (p), which converges similarly for values of p that are not infinitesimally near any of the roots of any of the equations ^{p + m) = 0, is at once derivable from its mode of con- struction from n(p). The row of quantities Xnt.j'lp) ^^ ^(p) f*^^ the same value of m is derived from the row of quantities i^m.^ in il (p) for that value of m, through multiplication of the latter by h^(p)(f>(m + p). Hence D(p)-a(p)nj,.(p)^(„,+p) where ii(p)=nK(p)<l-{^ + p) -Jt [i{(' * 'if"-') " " '""']] .i <" " "•'• and 11' implies multiplication for all values of m between + oo and — cc except m = 0. Also n(/')=^'^j;Msm(p-p.)'rl. Now D{p) has been proved to be finite (that is, to be not infinite) for all finite values of p ; and manifestly, from its form, it is a uniform funct.ion of p, so that it is a holomorphic function of p everywhere in the finite part of the plane. Further, D,{p) is a uniform function ofp; and it has been proved to be not infinite for values of p, which are not infinitesimally near any one of the roots of any of the equations (p(p + m) = 0, the aggregate of all these roots being pi + m, p^-i-m, .... pn + 'm, (i« = — oo to + oo ). Hence, owing to the relation D{p)-n{p)n(,p). y Google 117.] DETERMINANT 375 it follows that these roots are poles of ^(p)- Take a line in the p-plane inclined at a finite angle to the axis of real quantities, choosing the inclination so that it does not pass thi'ough any of the points p^ + m for all values of <r and m ; let it cut the axis of real quantities in a point f. Take the point /4-1 on that axis, and through it draw a line parallel to the former, thus selecting an infinite strip ia the /j-plane. Since a{p + i)^a(p), the uniform function il{p) undergoes all its variations in that strip: and within the strip, we have Lim fi (p) = 1. Owing to the nature of the poles of il (p), the strip contains n of them, which may be regarded as the irreducible poles : suppose that they are pi, p^, ...,/3«. Within the strip, p= oo is an ordinary point of the simpiy-periodic function ii (p) ; it follows* that the number of its irreducible zeros is also n, account of possible niuitiplicity being taken ; let these be /j/, />/, ..., p«'. Hence n(p) = A ^^^ t(P - Pi') '^j si n {(P - PaO ' ^1 - ■ ■ sin {( p - Pn) ■^] sin i(p - pi ) wj sin Kp - p, ) tt) ... sin Kp - p„ ) tt) ' taking account of the holomorphic character of -D(p) for finite values of p, and of the relation Here, A is independent of p. To determine A, we use the property Liran(p) = l, which holds for p = u+ iv, in the Hmit when v is infinite, whether positive or negative. Taking v positive and infinite, we have and taking v negative and infinite, we have yGoosle 376 FORMATION OF [117. Hence XpJ — Sp„ is an integer ; if it is not zero, we can make it zero by substituting, for the quantities p',, values congruoiit with them. Assuming this done, we have 2;>/ = Sp„, ^ = 1, so that ilip)'- and therefore Moreover, the quantities pt, p-i, ■■■, pn a^e the roots ol ^{p)~ 0, ao that hence Sp.'-i»(«-l). 118. Next, we consider the expression »in((p- -Pi. |T| |...sin((p- -Pn )■'] sinUp- -Pi" )ir\ |...sin(((>- ■Pn M' DM- : n .h. iKpj-i'W =IQ-^ i to prove that this series converges for all values of z within the annulus. It manifestly arises from D (p), on replacing ;\;o fc in D (p) by 3* ; we shall therefore assume that F is transformed into this modified shape of i)(p). When the determinant is in this shape, we multiply the column associated with m by z~™, and the row associated with m by z™] these operations, combined, do not change %m,ra, and they do not alter the value of the determ- inant. Let this combined pair of operations be carried out for all the values of m from —x to + co ; the result is to give a determinant, which is equal to Y and has Xv,,"~' for its constituent in the same place that ;^j,^, occupies in 1) {p). Hence, as for D (p), so T converges uniformly and uncondition- ally for values of p within the p-region selected, and uniformly and unconditionally for values of z within the annuius, if the doubly- in finite series y Google 118.] AN INTEOBAL 377 converges uniformly and unconditionally within those regions, and if S(x«-1) converges uniformly and unconditionally. The latter condition is known to be satisfied, owing fco the convergence of J) (p). It remains therefore to consider the con- vergence of the double series. With the notation of §§ 115 — 117, we have Now A^ (\) s' = a;„_^,. C;, x_5 3^ + a„_3, ,_, c,, x_, j^ + ...+ Cr. w sh- owing to the definition of the coefficients in the original differen- tial equation, the series converges uniformly and unconditionally, for values of a within the annul us R<\2\<B'; and therefore the series converges uniibrmly and unconditionally for the same range. Denoting this by J^, we have and \Jf\ is not infinite for any of the values of z. Again, as (§ 117) and when m is not zero, we have ■*.W%r^^ y Google 378 CONSTRUCTION OF [118. Proceeding with the double series SSxni.M-^'""^ exactly as with the double series £x"i.f > omitting for the preaent the terms corre- sponding to m = 0, and remembering that the summation is for all values of m other than m =/i, we have <K^J every group of terms in which is finite, so that SS|x,,,^»-| is finita Also, taking account of the terms omitted for the value m — 0, we have S |c.,,^'| < \J,: If'-", + i./.| |p"-; + ... + IJ-J, which is finite, so that summed for all values of m and ^ between — oc and + co except m = /i, converges unconditionally. Moreover, all the series which occur in the supenor hmits in the inequalities converge uniformly, both for the values of s considered and the retained range of p ; hence the double series converges uniformly and unconditionally. The proposition is therefore established for £©"■ A similar investigation shews that the series for any value of r, the numbers a and /S being any whatever, converges uniformly and unconditionally for values of z within the annulus, and for values of p in the range that has been retained. y Google 119.] CONSTBUCTION OF IrREGULAE INTEGRALS. 119. These results may now be used, by a generalisation of the method of Frobenius in Chapter iii, to construct expressions for the integrals of the equation Writing 1/= 2 (I™ 3''+'^, and adopting the notation of § 115, we have -e,(p) «»+'-», it GmW-O, for all values of m between — oo and + oo , except m—i. The last equations are equivalent to /i„<p)e»(p)-o, that is, to for all the values 0, + 1, + 2, ... of m, except m, = i. Let We have that is, for all the values of h. Hence, writing o...Af'V we have and y Google 380 IRREGULAR [119. Thus the quantity y, where ^-^-!©^'" satisfies the equation The determinant D (p) is of normal form ; the series for y con- verges uniformly and unconditionally, alike for vahica of s within the annulus R<\z\<R', and for values of p within the finite region contemplated. 120. Let p=p' be an irreducible simple root of D(/j) = 0. Then the first minors of constituents in any line cannot vanish simultaneously for p — p'', for and the left-hand side does not vanish for p = p'. Selecting minors of constituents in the line i, we havo and that is, ^1 13 an integral of the equation. Similarly for any other irreducible simple root of D {p) — 0. 121. Next, let p = p' be an in-educible multiple root of i> (p) = of multiphcity a. Firstly, suppose that some of the first minors of D(p) do not vanish for p = p'', let some of these non-vanishing minors be minors of constituents in the line i. Then, in the vicinity of p = p, we have as a quantity satisfying the equation P (2/) = ^sM-i-" (p - p-f R(p- p'% where Rip — p") does not vanish when p — p. It therefore follows that ^^-ip-p'y-'^^i^'P'P')' yGoosle 121.] INTEGRALS SO that, if ^< 0" — 1, we have \dp''),~/ is an integral of the equation. Hence, corresponding to the irreducible root p' of multiplicity a, there are integrals ^'^^\h 0] ^'^'' "^ ^" ^"^ ^ = ''^ + ?/« i«g ^^ 3/. = 2 r|^, Ql zf-^" + 2^, log ^ + y, (log s)= = 9jj + 2iji log s + y, (log ^)^ y.-.^7),_i + (^-l)7,,„,log^ + ^- ''~y^^^ -"^^^.^(log^)' + ... ... +(<r- l)j?,(log3)^-' + yoaog^)'"'. when, in each of these expressions on the right-hand side, we take p = p. 122. Next, still taking p = p' to be an irreducible root of i> (p) = of multiplicity a, suppose that, of the minors of successive orders, those of order r are the first set which do not all vanish for p — p'. Let the lowest multiplicity of p' for first minors be o-,, for second minora be o-^, and so on up to minors of order r — X, the lowest multiplicity for which is denoted by a-,-,. Then, owing to the composition of B in relation to first minors, to the composition of first minors in relation to second minors, and so on, we have <r><ri>(r,>...>o-^,. There are two ways of proceeding, according as r < ff, or r = a-. First, let r < cr. With the preceding notation, wo have and y Google 382 SUB-GROUPS OF [122. After the explanations given in the construction of these expres- sions, we know that p= p is a root of m.ultiplicity o-i for some of the minors in the expression for y. As before, in § 121, the quantities dy 3^ ^' dp' ■■•■ dp--'' when in each of these we take p — p', are such that for A, = 0, 1, ..., <r — 1. But owing to the fact that p = p' ys, a root of all the minni-s I ,1 of multiplicity ctj, all the quantities dy d-^y ■'' dp' ■■■■ dp-'-' vanish when p — p'- Hence the n on -evanescent integrals which survive are dp"' ' 3p''i+^ ' ""' 3,3'^' ' when p = p'. They have the form J,,. = aI^ Q «'« + (a, + 2) ,„ log . and so on : their number being Next, p — p' is, a root of least multiplicity o-i for some of the minors of the constituents of any line i: and there mast be at least two such minors. For i-w-sQ-v y Google 122.] IRREGULAR INTEQBALR §83 if p = p' 13 & root of multiplicity o-j + 1 for all the minors but [ , ] , then, as it is of multiplicity ^ .Ti + 1 for Z* (p), it would be of multiplicity o"! + 1 for f J. Similarly for any other line. Once more substituting in P(y), we have = t?i (p) z'-+'-" + Gj (p) z<-+^-^, provided G^ {p) — 0, for all integer values of p from -co to +x except p — i, p=j. The last equations are equivalent to kp(p}G^(p)=0, that is, to for all integer values of p except t and j. Consider quantities ag of the form for ail values of 0, tho quantities A and B being arbitrary. With these expressions for a^, we have Each of the sums on the right-hand sides vanishes, when p is not equal to either i or j : and thus the preceding expressions satisfy the equations «,((>) e,(p)-o, for all integer values of p except i and j. Further, *,(p) ft w ='?»,,«, y Google 384 SUB-GROUPS OF [122. and, similarly, i,(rte,(rt=-^(;)+i)Q. Using these values, we have as the expression for y ; and it satisfies the relation i* iy) = Gi (p) ^"^"" + Gj (p) 3^+j-» As the right-hand side of the last equation has p^p' as a root of multiplicity a-j, the quantities hi(p) and hj(p} having no zero for finite values of p, it follows that -(PI.-' for X = 0, 1, ..., ffj — 1. Therefore all the quantities dy d"'-'y when p = p', satisfy the equation P (w) - 0. Owing to the form of y above obtained, which has p = p' rr a root of multiplicity <rj, all the quantities ^' dp' ■■■' dp"'-' vanish when p — p. Therefore the surviving in1 '"-if- ,d so on : their number being y Google 122.] IRREGULAR INTEGRALS 385 Similarly for the next sub-group. With the same notation as before, we have P (y) - G, ((>) ^'+'-" + B,W «'+'■- + a. (p) ^■+»-", provided for all values of^, other than i,j, h, from — « to « . The analogy of the preceding case suggests for all values of 6, where A, B, C are any quantities. With these expressions for oe, we have =AX Each of the three sums on the right-hand side vanishes, when p is not equal to either i or j or k: so that the preceding expressions for a# satisfy the equation e,(/>)-o, for all values of p other than i or j or h. Further, Thus where ^(e, p) is a linear combination of minors of the second order; and the coefficients a^ being linear combinations of third order. y Google 386 SUB-GROUPS [122. As *(2, p) has p=p' as a root of multiplicity a^, it follows that for X = 0, 1, ..., C72 - 1 ; HO that all the quantities ^' dp- ■■■' ap"-' when p — p', satisfy the equation P (w) = 0. Owing to the form of the coefficients a^ in y, each of which has p^p' as a root of multiplicity a-^, all the quantities dy d"_^y ^' dp' ■■■' dp"-'' vanish when p~p' \ and we therefore are left with the integi-als and so on : their luiniher being Proceeding in this manner, we obtain successive sub-groups of integrals ; the total number in the whole group is which ia the multiplicity oi p = p' as & root of D (p) = 0. 123. Two cases, both limiting, call for special mention. It is manifest that, if tr — o-i > 1, the first sub-group contains integrals whose expressions involve logarithms; likewise for the second sub-group, if ffi — (ra>l; and so on. If, then, all the integrals belonging to the multiple root p = p' of Z* (p) = are to be free from logarithms, we must have 0-- 0-1 = 1, o-j-o-,-1, .... yGoosle 123.] OF INTEGRALS 387 and there fort! which thus is a limiting case of the preceding investigation. An intimation was given that, when r = o", a ditiferent method of proceeding is possible. As a matter of fact, the property of the infinite system of lineHi relations, ostablished in § 113, leads at once to the result. Let be one of the non-vanishing minors of order r belonging to D (p) ; then and the quantities a.^^, a^^, ,.., a^^ are bound by no relations, so that they are arbitrary constants. The integral determined by these coefficients is it manifestly is a linear combination, with arbitrary coefficients «■>.,, •■■- C'Mr, oi r integrals which are, in fact, the group of integrals above indicated. The other limiting o;\se occurs when r = 1 : all the <r integrals belong to a single sub-group. In that case, there exists at least one minor of the first order which does not vanish when p=p'; the condition is both necessary and sufficient. 124. We thus have a set of rr integrals, belonging to an irreducible root p' of D(p) = which is of multiplicity o-. Similarly for any other irreducible root of D{p) = 0; hence, when all the irreducible roots are taken, we have a system of n integrals. We proceed to prove that this system of integrals is fundamental. For, in the first place, it follows (from the lemma in § 27) that the integrals in any sub-group are linearly independent, on account of the powers of log^ which they contain. Next, there can be no relation of the form Ci2/.. 1 + O^y^ , + ... + C^yr.i = 0, yGoosle 388 FUNDAMENTAL SYSTEM [124. with non- vanishing coefficients 0. If such an one could exist, the coefficient of every power of z in the aggregate expression on the left-hand side must vanish. Writing i, j, h, ... =Pi, Pi, Pi, ■■-, Pr\ k, I, m, ... =5,, q„ q„ ..., q^ ' we have "-M'e. 'PuP2,--;pA , (Pl,p2, 'P„P^, and the quantities Ag,, A^^„ ..., j4s.s f're at our disposal. Let tiiese last be chosen so that Then the coefficient of sf'+'i in ^s,i is zero if (<s, and it is different from zero if i = s: let it be denoted by [ye,,\- The above relation being supposed to hold, select the co- efficients of 3^'+^', 2*'+?', .... z*''+9r in turn. As they vanish, we have 0. b.,i + c. [*.]., + ... + c, [».,]„ = 0, from the coeSicient of sf'+^^ ; every terra vanishes except the first, and [yi.i\, is not zero ; hence Cj = 0. The vanishing of the coefficient of if'*^' then gives 0. [.»., 4, + c. b.,.],. + . . . + C, [,j,, ,],_ = ; every term after the first vanishes, and [ya.ijg, does not vanish; hence And so on ; every one of the coefficients G vanishes ; and thus no relation of the form yGoosle 124.] OF INTBGEAI^ 389 Next, there can be no linear relation among the o- membei-s of a group. For, in any expression SO,,j,„ the coefficient of the highest power of log z is of the form and this can vanish, only if the coefficients Gs_i are evanescent; hence 'ZG^tys.t can vanish, only if the cocfiicients Gs_t a-i'e evan- escent. Lastly, there can be no linear relation among the members of different groups. For let Y{p',z), Y{p",z),... denote the most general integrals of the groups belonging to the irreducible roots p', p", . . . respectively, of D (p) = 0. Let z describe a contour enclosing the origin; then Y(p', s) acquires a factor e'""^, Y(p", z) acquires a foctor e^^", and so on. Thus, if there were a relation aY{p\s)+0Y(p',z) + ... = Q, then ae«">' Y(fi', z) + ^e="*" Yip", z)+...^0; and similarly, after k descriptions of the contour, ae^-ip'' Y {p, z) + /Se=^^"" Y (p", s) + . . . = 0, for as many values of the integer k as we please. Now p', p", ... are the irreducible roots of D (p) = ; no two of them are equal, and no two can differ by an integer. Hence the preceding rela- tions can be satisfied, only if in other words, no linear relation among the n integrals can exist. They therefore form a fundamental system. The Equation D(p) = is the Fundamental Equation of THE Singularity. 125. Consider the effect which the description of a closed contour, round the origin and lying wholly in the annulus, exercises upon this fundamental system. Let y Google 390 FUNDAMENTAL EQUATION [125. and let y' denote, at the completion of the contour, the value of the integral which initially is y. We have and so r where Hence the fun(! a mental equation {Chap, ii) is @ = 0, &~e. , , ■ , a.i , ef-e. , . ., . a^i , e'-e, . , , , , . ., A, , 0,0,. «'-», 0, . , ... ., , .. ■ , , .. , 0,0,. 0,0,. -«, .. where a' is the number of integrals in the group belonging to the root p of D{p)=0 of multiplicity a' ; a-" is the number in the group belonging to the root p" ; and so on. Now it was proved that Dip). n I ■Kc-p.')"!. if Henof sin (p - p.') ,r = g « - 1' + '.') " (.2"> - e'-' ^^1"^ - (ZSji'""'"""^''','?/''" '^'■ y Google 125.] OF THE SINGULARITY Also (§ 117) lp„' = in(n-l), so that g-T'2p„'=,g-in(«-l)Ti^ + 1; and therefore Dip)- As the quantity e""*"^' has no zero for finite values of p, it thus appears that, so far as roots are concerned, 2> (p) = and = are effectively the same equation, when the relation between p and d is taken into account. Also, so far as roots are concerned, il{p)=0 ia effectively the same as D(p)=0; hence ant/ one of the three equations = 0, D(p) = Q. n(p) = 0, may be used for the determination of 6 and the associated quantity p. It is known that = is ao equation remaining invariaative for all raodiiications of the fundamental system : and, for the form of equation adopted in § 114, the term in independent of is equal to unity (§ 14), This property in the present cose is verified by means of the values of the quantities 8', 0", ...; for (_^y(_^Y'... = (-l)''e^'''^''"' = (-lf e"l"-^J^' = (- 1)". The remaining coefficients in are known (§ 14) to be the in- variarUs of the equation, whatever fundamental system be chosen. Replacing by ii (p) for purposes of this discussion, we have ^ si n {(p - p/) tt] ... sin {(p - p/) 7r| ^^' sin|(p-p,)7rl...sin|(p-p„),r!- I^ow where so that, as and therefore sinl(p-p,)»l »-«,■ ipr - ip;, y Google 392 FUNDAMENTAL [125. we have ^P' (<)-e,)(e-8,)...(0~en) ^_. .. + (_!)«■ Hence, when il(p) is expanded in descending powers of d, the term in 8" is unity ; and when it is expanded in ascending powers of 6, the term in 0" is likewise unity. When the quantities &,, 0^, ..., 6^ are unequal, then Xi(p) can be expressed in the form On account of the character of il (p), when expanded in ascending powers of 8, we have so that there are w — 1 independent quantities Jl/,', and these are equivalent to the w - 1 invariants. The equation may also be expressed in the form n{p) = l + t M, cot [{p - p„) tt], where and therefore i #,= 0. Corresponding expansions occur in the case when equalities occur among the quantities p^, p^, ..., pn- 126. The integrals, which have been obtained, are valid within the annulus represented by R^\s\^R'; the inner circle may enclose any number of singularities of the equation, and the outer circle may exclude any number of other singularities of the equa- tion. But care must be exercised in particular cases. If for instance, the only singularity within the inner circle is the origin, and the integrals are regular in the vicinity of the origin, then in the expression of any integral, such as yGoosle 126.] EQUATION 393 there can be only a finite number of terms with negative values of m : the method, which is baaed upon the supposed existence of an unlimited number of such terms, is no longer applicable. If the only singularity outside the outer circle is z — <xi, and if the integrals are regular in the vicinity of 2 = cc , then in the expres- sion of any integral, such as there can be only a finite number of terms with positive values of m. : the method again ceases to be applicable. In 3u<ih cases, the best procedure ia to construct a fundamental system which shall include the regular integrals : this ia the customary procedure for, e.g., Bessel's equation, the integrals of whicli have 3= co for an essential singularity and are regular near s = 0. The method, which uses infinite determinants, is best reserved for equations which have their integrals non-regular in the vicinity of every singularity : it is nugatory when applied to Bessel's equation. Ex. 1. Consider the equa-tio S+S+ b-"- It is clear that the point 3 = is an esBential singularity, there being i integral regular in its vicinity, when a is different from 0: and that 2=co likewise an essential singularity, when y is different from 0. "We ahall aasun that both o and y arc non- vanishing quantities. Let the eqiiation becomes dhi (a h a\ With the notation of the preceding paragraphs, we have 4.(p)=p(p-l)-l-6=(p-p,)(p-pj); Or,g,=(i, when /t<!'- 1, and when fi> r+l ; ^,,^=0, when /i<r- 1, and when >i>»-+I. y Google 394 EXAMPLES Henco the value of n (p) is [126. ..., 0, *(p"-"2) 1 ' *(p-2 f " , , 0,... ..., 0, 0(p-l ' ' f(p-i) <> ,0,,.. ,.., 0, ' -Hp) 1 1 w , 0,... -, 0, ^(H^i) 1 ' *(P + 1) , 0,,.. ..., 0, " LI , *(p + 2 ' *(7+2)' "'■■■ The general investigation ahewa that, wheu p, and p, are unequal (which will he assumed), Q(p) = l+ifi7rcot{p-pi)n- + J/3^cot{p-pj)jr, with the condition that is, we have Q(p)=l + 7rJ/[coti(p-p,)ff}-Mt{(p-p2),r}], where M is itidepeiidcnt of p. Taking the determinantal form for Q (p), and espanding according to the law established in § 110, we have where odd powers of a do not occur because the combinations which they multiply all vanish. Also , ™+i0{p+m)0(p+m.+l)^(p+?))<^{()+F+l)' 2 2 ^(p + m)>(p + ™ + l)^(p+p)<^(p+p+l)<fi(p+g)0(^ + 5 + l)' Hence we have To find jVj, we notice that the only terms in M^, which have p = pi for a pole, are those given by m=0, m= - 1, these being <^(»^(p+l) 0(p-l)^(p)" yGoosle 126.] hence <j>{p)={l>-p,){p-p2)\ P,-pA<Pipi+V 0(pi-i)J Pi-Pa l-ipi-p^y ^^ipi-pi) Again, writing $(p + m) = ^(p + ra)(>(p + Mi + l), we have ^*=^^*{p + m)*{p+p)- Consider ij_„*(p+«)}' it contains the terms ik^J" which do not oc ;cur in i/^,; it cont-vins terras which do not oc icur in J/, ; and it contains the terras 1 ^>«+i*{p + '«)*(p+p) twice over, onci .in the form and once iTi the form I Hence »>^+i$(p + m)*(p+^)' {?„*(p+7t)) J-A*{p + r>:)i '-.".*(p + ™)*{p4 -»-n,*^''. 80 that ^*=*{1I t{p+n)\ ^„J'-. l*{p + «)) -„*{p + ™. 1 )1.(p+m + l) The first term ■ on the right-hand side is = i.V3i'^2[cot(p-pi)^^COt(p-p,),r]S; the residue of this function for p = p, is = -lS\^-,vQat(p^-p^)^ -7rC0t{(pj-pj)«-} y Google 396 EXAMPLES [126. In the second tci'ni on the right-hand side, ttie residue for p=pi can arise only for the values n = 0,n= — I; thus it is , (l+2p|-2pa)(2 + p,-p,) , (l-2pi + 2pa)(2-Pi + P3 ) ' {Pi-Pi)Hi+Pi-Pif ^ (Pi-P3)={l-Pi + P2)' after reduction. Similarly, from the third term, the residue is 3-8?> l-8i 86«(3 + 4i)(p,-p,) Sbm-ih)(p,-p,)- Hence .r^_w_cot{(pi-p2)^)_ 3-H6-526'^ + lfii3 * 1662(1-46) " 86^{3 + 46)(l-4i}(pi-p5)' after reduction. Other coefficients could be calculated in a similar manner : but it is clear that even iVg would involve considerable numerical calculations, and it is difficult to see how the general term could thus be obtaiued. But the method of approsimation may be effective in particular applications. Thiis, in Hill's discussion* of the motion of the lunar perigee, the convergence is very rapid; and comparatively few terms need be taken in order to obtain an approximation of advanced accuracy. When this is the case, the values of p' for the integrals arc given by Q(p)=0, cos2p^=-cos{(p.-p,)^}-2^Jfsin{(pi-ps)^h and two irreducible values of p chosen are to be such that Pi'+P2'=pi-I-P2 = l- The expressions for the integrals are to be obtained. Denoting still by p either of the quantities p,' and pj', the relations between the coefficients are *oS:7)"'->+°'+.),(,T^'^*'-°- and considering in particular the row 0, we know that the constants a are propoi'tiona! to the minors of the constituents in that row iu the determinant n (p). Thus for ail positive and negative values of k : so that, if we take imoir already quoted ii y Google 126.] and our solution h. for the effective expression of which, it is sufBcient to find the first minors, as the series is known to converge within the annulus. I« «rd» to obt.d„ Q torn aO.), we npl,«. ^^^, ^"^-^y ^j^^ by zeros ; it will therefore be necessary to do this in the e We ttus have Q=^l + a-'M^,,+ a'M^, + ..., P+J:+S_ <p(p)'p(_p + l) 0(p-l)0(p) Sinailarly for M^ f from M^ ; and so on. In order to obtain [ , j from ii (p), we replace -— — ^ in the — 1 column \ - V 9 ip) by unity; the quantities 1 and -j-f—~i,\ in that column by zeros; and the quantities 1 and -■ . in the line by zero. Wo then easily find \-ij 4>{p-i) '"'^^(p)<t,{p+i)^<!>{p-i)<i,(py<t.{p-i)<i>{p-2) <p{p-l)4<{p-2} aud .similai'ly for the others. In the same way, we have (i)-*iVi)+-*"«*^'..'+-- ^■f.(p + l)^(p + 2)' and so for the others. y Google 398 MODES OF CONSTRUCTING [126. Lastly, for iiogfttive values of p Icaa than - 1 and for positive values gi-eater than -l-l, we have {^\==a-^M^,,^-\-a^M„,^-V .. where M, = Vn , H — — H and iu fti the tlei'! Alter the remarks made m lalatioa to the foimal development of Q (p) it 19 manifest thnt these expressitna for the integrals are mainly useful foi apprciimate numerical exparsions they cannot it ptesent lo held ti constitute a complete formulation of the mtegiib El 2 In his classical memoir alieidj quote 1 H 11 c3naiic.n> tie h fli h Kg US erab mailer than «, The mi^inoir WW ensaiof ean (jCLOimt bemg takpn of the la una, nrgn pLd yPu'ar) and for the numern,il approxima ns It will bo noticed that the eflectiveness of the method is lai^ly influenced by the data as to the smallness of a^, o^, .... when compared with a^. Ex. 3. Disciaa the equation in Ex. I, when 6 = J, so that px — H- Ex. 4. Given an infinite system of differential equations of the form lit :J^a^,„:r„, (™^1, 2, ..„«>}, where the coefGcients o^^ are regular functions of t within a region \t\ ^ R, such that ]«„,„! <S™,d„ in this region, where S^, J„ (for ra, w=l, ...,<») are such that the series S,^, + S2^24-...+(S„j1„ + ... converges. Shew that, if a set of Constanta e^, Cj, ... be chosen, so that the series Cl.d, + C2.d2+...+C„-^„+... converges absolutely, then a system of integrals of the equations is uniquely determined by the cnndition that a',„ = (^,„, when ( = 0, for all values of ra. (von Koch.) Other Modes of constructing the Fundamental Equation FOR Irregular Integrals. 127. The preceding nnethod, so far as it is completed, leads to the determination of the fundamental equation for a closed circuit round the origin, the circuit lying entirely in the annulus; y Google 127.J THE FUNDAMENTAL EQUATION 399 and it leads also to the determination of the integrals. Other methods have been proposed by Fuchs*, Hamburger ■(■, Poincar^|, and Mittag-Leffler|, some of them referring solely to the con- struction of the fundamental equation. But all of them seem less direct than the preceding method, due to Hill and von Koch ; and they are not less devoid of difficulties in the construction of the complete formal expression of the integrals. Ex, \. A modification of Hambui^er's method, applied to the equation abeady discussed in Ex. 1, § 126, may give some indication of his process. Changing the variable from ^ io t, where the equation!] i°'' ^ '' where df c=6-i. Let X describe a circle round the origin, say of radius unity ; then on the completion of the circle, t has increased its value by 2ir. Let y=f{^), y=3 (a^) be two linearly independent integrals ; and when x describes its circle, let these become [/(if)], \ff (^)], respectively, so tliat [/(^)] = «,./(^)+<tisS'W, [6'(^)] = %/(^)+«22ffW- The fundamental equation for the circuit is * CrelU, t. Lxsv (1873), pp. 177—223. + Crelle, t. Lissin (1877), pp. 185—209. In oouneotion with this r reference shoulil be made to two papers by Giinther, Crelle, t. cvi (1890), p[ 336, ib.. t. cvri (1891), pp. 298—318. X Acta Math., t. iv (1884), pp. 201—319. In connection with tliis i reference should be made to Vogt, Ann. de VEc. Norm., SSr. 3% t. vi (1689), pp. 3—71. 5 Ada Math., t. sv (1891), pp. 1—33. II In this form, it is a spcoial case of Hill's equation : nee Es. 2, g 126. y Google 400 EXAMPLE [127. that is, by Poincare's theorem {§ 14), <"^-(l'll+«22)'» + l=0, so that Ci, +<i22 is the one invariant for the circuit. Let The fundamental equation is indepondejit of the choice of the linearly independent system, and it is unchanged when any particular selection is made. Accordingly, let the integrals be chosen so that F(i) = l, F'{t)=0, G{t) = 0, O'W-l, when ( = 0; then, using the foregoing equations, we have /■(2,).-«,„ ff'(2,).-o„; and therefore which accordingly gives the value of the invariant, when the values of i''(27r) and G' (27r) are known. To obtain these, let so that a increases from to 1, as ( increases from to Stt. The equation becomes and this remains unaltered when we change u into 1 - u. Two linearly independent integrals, constituting a fundamental system in the vicinity of M = 0, are given by where a|,=l, Cii=l ; also «„ is the value of 6„ when p=0, and c„ is the value of 6„ whet! p=i, the quantities 6„ being given by the equations (p + 2)(p + f)6a = !{p+l)2 + 4i^ + 8a}&,-64a, and, for values of )t^3, ()i + p)(!i + p-i)6„={(n + (>-l)'H4c + 8ai6„_i-64(i6,^j + 04o6„_3. Similarly, a fundamental system in the vicinity of u~\ is given by Z,- 2 «„(!-«)", Z,= S c„(l-«)"H. yGoosle 127.] EXAMPLE 401 Tho integral F{i), defined by the initial conditions /"(() = !, F'{t)^0, wlien i=0, is given by The integral f?[i), defined by the initial conditions i3(0 = 0. ^'[0 = 1. when t — 0, is ^b/^n by (;(<) = 4F2. To obtain expressions for Fi^n), Q' {'in), consider values of u, which lie in the vicinity of m=1 and are less than 1, By the ordinary theory of linear equations, we have First, let «=|, so that 1 — « = ^; then we have F{':r) = AF{^)+lBQ{n), a{^) = iCF{w)+DG{n). Next, differentiate with regard to a, and then take m=J, 1 — a=J; we have F- {^)^ - AF' M-iBG' {t^), 0'{n)=-^CF-{n)-DO'{n). Moreover, F(()G"(()-F'(0<?(0=constant = 1, by taking the initial values ; hence These rel^ttions give A~FM e'M+?"M (?(,). -A Hence e(2fl--T) = 4r2(2n--T) F{2,t~T)=Y^i2n-T) .AZ^{r)*BZ,{T), y Google 402 EXAMPLES [127. Now, when t is tt, the value of u is ^, so that and therefore ■.„+«,.--f'(2,)-e'(!,) — L.i!".i. 2-" .:.2— .ioH whieh 13 the inyariant of the fmidimeutil equation This gives a fiimal expreosion, the onU operatuno requiied lieing m the direct construction of i^ and c„, ■ind no one ot thene tperation"; k inverse but the result IS less fcuited to numem il ippiiiimation than la the method of infinite determinants in the case when a h small "We ah^ll return late! (§§ 137 -130) to t difterent diSCusBion of this equation. Ex. 2. Applj the preoeditig method to Hill's equation - -j^=ao+«iC08 2(+aaCos4(+..., in the case when %, a^, ... are not small compared with o^. Ea;. 3. Discuss, in the same manner as in Ex. 1, the equation In particular, obtain espressions for the invariants of the fundamentiil eqiuv- tion for z=0. y Google CHAPTEB IX. Equations with Uniform Periodic Coefficients. 128. Am, the equations which hitherto have been considered have had uniform functiona of the variable for the coefficients of the derivatives ; and the only particular class of uniform functions, that has been specially adopted with a view to detailed discussion of the properties of the equation, is constituted by those which are rational. Many of the properties, however, which have been established in the preceding chapters, hold for uniform functions whose form, in the vicinity of a singularity, is similar to that of a rational function when expressed as a power-series in such a vicinity. Among the classes of uniform functions, other than rational functions, there are two characterised by a set of specific properties ; viz. simply-periodic functions, and doubly-periodic functions ; and accordingly, it seems desirable to consider equa- tions having coefficients of this type. The present chapter will be devoted to the discussion of equations the coefficients in which are uniform periodic functions. Equations with Simply-periodic Coefficients. We begin with the case in which the coefficients have only a single period ; and we take the equation in the form where p,, ..., p^ are uniform functions of z, are periodic in w, and have no essential singularity for finite values of 3. Let a y Google 404; EQUATIONS HAVING [128. fundamental system of integrals in the domain of any point be denoted by /.W. /.W /-<4 which therefore are linearly independent. A change of 2 into z+tn leaves the differential equation unaltered : hence /.(2 + «), /,(^ + ») /,(^ + «) are integrals of the equation. That they are linearly independent, and therefore constitute a fundamental system (it may be in a new domain), is easily seen ; for satisfies the equation for aJl values of s, and by making s pass from any position Z+ a to Z without meeting any singularity, the integral changes from XcrfriZ + a) to ScrfriZ). If, then, values of c could be found such that the equation is satisfied identically (and not merely for special zeros of the function on the left-hand side), then we should have 2c,/.(^) = 0, also identically. The latter is impossible, because the integrals A{z), ■■■,fmi^) constitute a fundamental system; and therefore the former is impossible. Thus /, (a + ro), . . , , /,„ {e + oi) constitute a fundamental system. Suppose now that the domain, in which the original funda- mental system exists, and the domaan, in which the deduced fundamental system exists, have some region in common that is not infinitesimal; and consider the integrals within this common region. As fi{z-k-a), ..., /^(•^ + w) are integrals, and as/, (e), ...,f,a{z) are a fundamental system, we have equations of the form /, (^ + ») - o„,/, w + . . . + «.,/. w ) where the coefficients a are constants ; their determinant is not zero, because the set of integrals on the left-hand side constitutes a fundamental system. y Google 128.] UNIFORM PERIODIC COEFFICIENIS 405 Consider any other integral in this region ; it is of the form FW = «,/,W+.,/,W + ...+«,A,W, where /Ci, «2, ..,, «™ are constants; and so ii' (» + «) = 1 <!„«,/, W + I »„«,/, (2) + . .. + I »„«,/, W, In order that F(z) may be characterised by the property ^ constant, the coefficients k must be chosen so l<i^«r = ^«p, (;» = 1, 2, ...,m). where d i that a set of n equations linear and homogeneous in the coefficients k ; and therefore S must satisfy the equation (La , a^-d, ..., a^ an equation involving the coeSicients a, and so apparently depend- ing ixpon the choice of the fundamental system /i, .,., f^- But, as with the corresponding equation for a set of integrals near a singularity (| 14), we prove that this equoiion is independent of the choice of the fundamental system, so that the coefficients of the powers of 9 are invariants. The proof follows the hues of Hamburger's proof for the earlier proposition. Let another fundamental system g^{z), .... ffmi^)- existing in the region under consideration, be such that ?-(»■ ")-in».W + .- + f>™9..W. ('■-I.' «}. the determinant of the coefficients b being different from zero. The equation, to be satisfied by the multiplier of F(s), is B(S). ill - «, y Google 406 FUNDAMENTAL EQUATION [128. As the integrals / are a fundamental system in the region, in which the integrals g exist, we have 5',(3)-c,,/,(2) + ...+c,^/™(4 (5 = 1, ,..,m), where the determinant of the coefficients Cj;, say G, is not zero. Thus, as hn9, (^) + . . . + K^g^ = gr{^ + ») we have _l lb„c„f, (2) = l_ l^co., /, (,). This homogeneous linear relation among the linearly independent integralsy" must be an identity; and therefore S hraCst — S C^sffst say. Then = A(e)C, BO that, as C is not z we have B(0) = Aid), and the equation is invariantive. We therefore call it the fun mental equation for the period a. Let A (z) denote the determinant 4(.). '?::^^ ^^:^' ^":^ 1^9, we have in— da—' ■ ■ ■■ d.-' df, dz ' ■ df. ■■ di /. /. , . ., /- r "fA^)dx y Google 28.] >that FOB THE PERIOD A {« + «.) I'^'pil')" where we may assume the integration to take place along a path that does not approach infinitesimatly near the singularities of pi, if any. Now, as pi is a uniform function, simply-periodic in m, it is known* thatpi is expressible in the form within such a region as encloses the path of integration ; and the series is a converging series. But ly? if the integer a is distinct from zero ; hence aw " ■ But, substituting in A(s4-w) the expressions for /i (z + w), ..., /m(^ + f^} ^rid their derivatives, in terms oi fi{z), ..., /™(s) and their derivatives, we have A(» + ») which is the non-vanishing constant term in A (0) ; and thus In particular, when pi is zero, so that the differential equation , we have ^„ = ; and then contains no term in - A(0) = l^ K-ir^™- 129. The generic character of the integi-als depends upon the nature of the roots of the fundamental equation, y Google 408 ROOTS OF THE [129. If the m roots of the fundamental equation are different from one another, and if they are denoted hy ff^, 0^, ..., ^m, then a fundamental system of integrals exists, such that K(^+«)^e,F,(4. (---i '»)• Consider any simple root ffr of the equation A ($) = 0. Then not all the minors of A {d) of the first order can vanish for $ = ds-', hence m — 1 of the equations I a,pK,= eKp. (p = h 2, ..., m), determine ratios of the m quantities k, and consequently determ- ine a function i^r(^) having a multiplier ^,. This holds for each of the m different roots: and thus m different functions F{z) are determined. These m functions are linearly independent of one another. If there were an equation 7i^i (2) + ^,F,(s) + ...-]- j^F^ (^) = 0, which is satisfied identically, thea also y,F,(s+6y) + y,F,{z+m) + ... + y^F^{z + o,) = Q, that is, ^ijiF, {z) + d^jj\ (s) + ... + e^'i^F^ {z) = 0. Similarly, ^1^71 -fi («) + ^.'7i^. (s) + . .. + 0^'y^F^ (s) = ; and so on, up to d,"^'yiF,(s)+o,^-'y,F,{0) + ... + e«,-^-'r.>^F>n(s) = o. Now the determinant W, e,\ 6i, ..., e^'^'l does not vanish, because the quantities are unequal ; hence so that the constants 7 all vanish. The m functions F therefore constitute a fundamental system. 1.30. Next, let °r be a root of ^(^)=0 of multiplicity ^, where ^ > 1. The equations ^ a,ipKs = 0iCp, (i>= 1, ■■■> m). y Google 130.] FUNDAMENTAL EQUATION 409 are consistent with one another, though not necessarily inde- pendent of one another : any m — 1 of them are satisfied hy ratios of the quantities k, which are finite and may contain arbitrary elements. Giving any particular values to the last, we have an integral, say 'I>i {z), defined by means of these quantities : it is such that and it is a hncar combination of/, (^), ..., f^{z). Taking any one of the integrals which occur in the expression of this linear com- bination, say /i {z), we modify the fundamental system so as to replace f^ (s) hy "!>, (a). Let the equations for the increase of the argument by oi in the modified fundamental system be /.(^ + t«) = c„*,(^)+c./,(2)4 then the fundamental equation is fc,,„ /,„(.), (r 'h-6, which, owing to its invariantive character, is A {$) = 0, and therefore has S for a root of multiplicity /l. Consequently, the equation has ^ for a root of multiplicity ^ — t ; and therefore the equations (c^-'^)ic^'+c.^ K^' + .-. + o^/cJ =0, Cn^f^!' + 0^3*3' + ■ . ■ + (Cmm -^) K = 0, are consistent with one another, and any m — 2 are satisfied by ratios of the quantities «', which are finite and may contain arbitrary elements. Giving any particular values to the latter, and writing «>. W -«■'/. W + «.'/. W + .••+ «..7- W, we have ■D,(» + «) - x,,*, W + a*. W, y Google 410 FUNDAMENTAL SYSTEM [130. where so that Xsi is a constant, which may be zero. The quantity 0^{e) is an integral of the differential equation r we use it to replace some one of the integrals in its expression, say /^is), in the fundamental system, so that the latter then is constituted by s.W, i-.W/.W /~W- Proceeding similarly from stage to stage, we infer that, associated with a root ^ of multiplicity /j. of the fundamental equation, there exists a set of ft integrals such that 0, {^ + w) = V*. (z) + \,,*, (z) + ^*3 (2), where the coefficients X are constants. Similarly, if the roots of the equation ^ (^) = are %, ...,&„ of multiplicities ^u.,, .... n-a respectively, so that /^i + ... + fi,j^ — in, the fundamental system can be chosen so that it arranges itself in n sets, each set being associated with one root of the fundamental equation and having properties of the same nature as the set associated with the preceding root of multiplicity &. A function, characterised by the property is strictly periodic, and sometimes it is said to be periodic of the first kind. A function, characterised by the property F(, + „).ilF{z), where ^ is a constant different from unity, is pseu do -periodic, and sometimes it is said to be periodic of the second kind, 9 being called its multiplier. A function, characterised by the property where X and /j. are constants, is also pseudo- per iodic, and some- times it is said to be periodic of the third kind. y Google 130,] OF INTEGBALS 411 With these definitions, the preceding result can be enunciated as follows*:— A linear differential equation, the coefficients of which are simply-periodic in a period co, possesses integrals which are periodic of the secmid kind: and the number of such integrals is at least as great as the number of distinct roots of the funda- mental equation for the period. Ex. 1. Prove tliat, if the equation iPw , , .civ! , , , „ integral which is periodic of the third l^ind with a multiplier e"+^, then i.i(s+a,)=p,(s)-2X, Hence integrate the equation shewing that XiB = 4jr^. (Craig.) Ex. 2. Shew that, if the coefficients in the equation have the form p,(.) = 0(.) + ^, where i^ and yjr are periodic of the first kind, then the equation certainly possesses one integral that is periodic of the third kind. (Craig,) 131. On the basis of these properties, we can take one step towards the analytical expression of the integrals. The integral »I>, (s) is a periodic function of the second kind. As regards the integral '^^(s), we have *,(« + '«) ^A^) ■A' ' Floquet, Ann. de Vic. Norm., S^r, 2°, t. sii (1883), p. SS. that y Google 412 PERIODIC INTEGRALS [131. SO that the function on the right-hand side is a periodic function of the first liind, say t/t (z). Therefore where <I'isa (s) is a constant multiple of *i (g), and the constant factor may be zero ; and ^^ (z), = i/r (s) *i (z), is a periodic function of the second kind, with the same multiplier as *i (a). Vs regards the integral $3 (3), we have *, (s + ffl. <I>, (2 + 0. '.if -+w)-i -&"*<">■ _x,.x. , 2SW . + ^-^ 2&X. we hai 80 that ^(3) is periodic of the first kind. Hence where "t,, (a) = (z) ^^ (a), and therefore is a periodic function of the second kind with the same multiplier as "^ij where 'i':si(z) is a linear combination of ^^ (s) and ^, {z), and thus is periodic of the second kind with the same multiplier as ^i (s) ; and ^si{z) is a constant multiple of ^, (s), in which the constant factor, viz. may be zero, and certainly is zero if ^-aiz) disappears from (^^{s) owing to the vanishing of its constant factor. Proceeding in this way stage by stage, we obtain expressions for the integrals in succession ; and we find <^^ {Z) = •3>„ (S) + Z<^ri {Z) + 3= *rs (3) + . . . + 2'^'*rr (z), where ^ . (- ly X.,r-iXr-.,r- s ■ - X:.,X^i , . ■^"■W (r-lji^'M-- ''••'' 80 that it is a constant multiple of *i{^), the constant factor being capable of vanishing ; and all the functions 3>^i {z), ^^ (s), ..., *r,r-i(2) are periodic functions of the second kind with the y Google 131.] OF THE SECOND KIND 413 same multiplier as ^^ (z), and are expressible as linear combina- tious of O,, *,i, *3,, ..., <&r-i,i. This holds for the values r = 1, 2, ...,(^ Similarly for any other set of integrals, associated with any multiple root of the fundamental equation of the period. It may, however, happen that some one of the coefficients \,a^i vanishes, so that, for all values of r^s, the term in ^„ (a) disappears. The alternative result is that a linear combination of the functions ^g{z), ^s-i{z), ..., "I'lia) can be constructed which is periodic of the second kind. This linear combination can be used to replace ^s(^), and thus may be the initial member of another set of integrals in the group associated with the multiplier &. The proof of this statement is simple. Assume that \j,j_i vanishes, and that no one of the coefficients V,.., for yaluesofr^ a vanishes; and construct the linear combination choosing the coefficients k so that the term in "J^i disappears and that the remaining terms are ^ («,*, {z) + «,_,*,_, (2) + . . . + «,*, (z)\. To satisfy these conditions, we must have = «,\sl + K^lXs-1,1 + .■■ + K4V + fs!^3i + «3^, = KgXs.t-i + K^^'k^l,,-^. Transfer the terms in ics to the left-hand side : the determinant of the coefficients k on the remaining right-hand side is which by the initial hypothesis does not vanish. Some of the coefficients Xj,, Xgj, ,.., Xj,j_a are different from zero, for ^i{z) is not a periodic function of the second kind; hence there are finite non-zero values for the ratios of «s-i, ■■-, «= to «s- When these values are inserted, let *.{«)-«.*. W + -.- + «.'i>.Wi then y Google 414 SETS OF [13L so that 'Sfj (s) is periodic of the second kind, with the common multiplier ^ ; it can replace <E>s (z) in the fundamental sj^tem, and then can, like Oi(^), be the initial member of another set within the group of the same type as *i(«), ..., *«-i(s). The statement is thus proved. 132. Any set, such as ^, (s), ..., ^s-iCs) in the preceding group of integrals, whether s = /i or be less than /t, can be replaced by an equivalent set of simpler form. Let the equation be written BO that Also let P, SP P, 3-P and, generally. let P. d'P Let the integral of the set containing the highest power of z, say g''~', be expressed in the form ...4(r- l)«0^_, + ^„ the binomial factors being inserted for simplicity. Then, as F (Z'y}r) = 2-P (f ) + KZ'-'P, (l/r) + «(«- l)2-=Ps W + ■.■, we have = P(».) -^■P(« + ('--l)^'-P.(« + i('--l)Cr-2)i>"P,(« + ... + <r-l)K-'P (,(,,)+ (r - 2) z'-P, (« + ... + «r - 1) (r - 2) {^-P (« + ... y Google 132.] INTEGRALS which can be aatiefied identically, only if The first of these conditions shews that is an integral of the equation. The second shews that is an integral ; the third that is an integral. And generally, if w denote ^(M-l}!(r-/^)l^ ...+(7--l)f^^, + 0„ w being an integral of the equation, then each of the quantities _ _1 a^ 2!(r-3)! 5°w (fi-l)[{r-fj.)]3'-''w ^"^ r-iar (r-1)! ar^' "'■' (^-1)! S?*^"' ■" is an integral of the equation, when ^ is replaced by £ after differ- entiation. Accordingly, the group of r integrals in the set are linearly equivalent to i!^ = .^s + 301, 'ts = 03 4- 230a + 2^01 . M4 = 04 + 3^0, + 32=02 + ^'Vi , Ur = ^r + (r~ 1) 30^1 + .. . + (r - 1) 3'-^02 + 2-'<Pu and any linear combination of these is an integral of the differ- ential equation ; all the quantities which occur in them are periodic of the second kind, having the same multiplier. Similarly for any other set ; and thus the- vi integrals of the equation will he constituted hy sets of r^iVi, ..., r„ integrals of the - m, and the system contains riodic functions of the second kind. y Google 416 group of istegrai£ associated [133. Group of Integrals associated with a Multiple Eoot OF THE Fundamental Equation of the Period. 133. These results can aiso be obtained by using the proper- ties of the elementary divisors of the quantity A (0), when it is expressed in its determinantal form. Let the elementary divisors associated with the root S be so that, as in § 15, the highest power of ^ — ^i common to all the first minors of A {$) is (d — 'Sr)"^, the highest power common to all the second minora of A (6) is {d — ^)'^, and so on; and the minors of order t (and therefore of degree m — t in the coefficients) of A (d) are the earliest in successively increasing orders not to vanish simultaneously when 6 = '^. As in the earlier case dis- cussed in §1 15, 16, we have Proceeding on lines precisely similar to those followed in | 23 for the arrangement, in sub-groups, of the group of integrals aasociated with a multiple root of the fundamental equation belonging to the singularity, we obtain a corresponding result in the present case, as follows : — The group of ft mtegrah aasociated with the root ^ of multipli- city n,belonging to the funda/mental equation for the period at, can be arranged in r sub-groups, where t is the numh&r of elementary divisors of A {d) which are powers of $~^. If the X members of any owe of these sub-groups be denoted by gi{z), g^iz), ..., gt.{z), these irdegrals of the differential equation satisfy the characteristic equations J. (^ + »)-as,w •, jr. (2 + »)=.*» W+ ft W I j.(«+")=aftW+».(») r Taking all these sub-groups together, the number of first equations which occur in them is equal to the number of the sah-groups, that is, the number of the elementary divisors of A {6) connected y Google 133.] WITH A MULTIPLE ROOT 417 with ^ — 9- ; the number of second equations which occur is the same as the number of those indices of the elementary divis&rs connected with 6 — "^ that are not less than 2 ; the number of third equations is the same as the number of those indices that are not less tha/n 3 ; and so on, the number of equations in the first sub-group being The analogy with the Hamburger sub-groups in Chapter ii is complete. Corollary. The total number of integrals of the second kind, defined as satisfying a relation of the form g {! + «) = eg (.), where 6 is a constant, is the total number of elementary divisors of A(0) associated with all the roots of A(d) = 0; a theorem more exact than Floquet's (§ 130). For the total number of such integrals, in the group associated with a multiple root of A ($) = 0, is equal to the number of elementary divisors of A (ff) associated with that root : and the total number of groups is equal to the number of distinct roots of A (8) = 0. 134. Some approach to the analytical expressions of the functions, satisfying the equations characteristic of the sub-group, can be made, as in § 23. Let and introduce a difference -symbol V, such that* for any function F; also let /X-l\ ^ f\-l\ G <^) = x^ + (\') ?;..-. + (^ 2 ')rxA-. + ... where the functions j(i, ^2, ■-■, %>. are periodic functions of 3, with a period ro, and V r J rl(\-l-r)': * For theae difference -symbols in general, i Mat., See. 2", t. x (1882), pp. 10—45. y Google 418 GROUP OF INTEGRALS ASSOCIATED [134. Then if we take for a!! values of m, we have holding for all values of n. These are the characteristic equations of the sub-gi^oup ; and we therefore can write with the above notations, for n = 0, 1, . . X — 1 These \ integrals are a linearly independent ''ei out of the fundamental system; the system will remain fundunental if ffi! 9i> ---.^A *re replaced by X other functions Imeailj equivalent to them and linearly independent of one inothe-r This modifica- tion can he effected in the same way as the corresponding modifi- cation was effected in | 24, viz. by introducing a set of functions, associated with G and defined by the relations the functions '^^ being periodic functions of z, with tho period w. Constructing the expressions Vff, V^G, ..., V'-^G, we find V^-^G = cx_,,,Gi, where the constants c are no n- vanishing numbers, the exact values of which are not needed for the present purpose. y Google 134.] WITH A MULTIPLE ROOT 419 It follows, from the last of the equations, that G^ is a constant multiple of V*~'{?, and therefore that ^" G, is a constant multiple of ffi (s) ; we replace y, (s) in tlie fundamental system by ^ G^. It follows, from the last two of the equations, that G^ is a linear combination of V—^G and V'~^G, and therefore that S^"(?s is a linear combination of g^ (z) and g-i (s). As g^ {z) has been replaced in the fundamental system, we now replace gi{z) by ^"Gj; and the system remains fundamental. And so on, for the integrals in succession. Proceeding thus, we obtain X, integrals of the form ?ri9.(2), ^" (?,(«), ...,S-iS;^(s). Further, these integrals are linearly independent, and so they are linearly eqiiivalent to ^,(3), ^2(2), ...,gx(z). For if any relation, linear and homogeneous among these quantities, were to exist with non-vanishing coefficients, we should, on substitution for Gu G„ ..,, (?;vin terms of G',VG,V=G',.. .,V*-G', obtain a relation, linear and homogeneous among the quantities gi{z), ..., g^i^) with non-vanishing coefficients. Such a relation does not exist. Accordingly, the X integrals can be taken as constituting the required sub-group of integrals. We now are in a position to enunciate the following result, defining the group of integrals associated with a multiple root ^ of the fundamental equation of the period : — When a root ^ 0/ the fundamental equation A (0} = O is of midtiplidty fi, there is a group of fj, integrals associated with that root; the group can be arranged in a number of sub-groups, their number being equal to the number of elementary divisors of A {$) which are powers of "it — 6 ; the number of integrals in the fi/rst sub-group is equal to the number of those elementary divisors ; the nwmber in the second sub-group is eqtial to the number of the exponents of those divisors which are equal to or greater than 2 ; the number in the third sub-group is equal to the number of the 27—2 yGoosle 420 GROUP OF INTEGRALS [134. eicpan&ixts of those divisors which are equal to or greater than 3 ; and a suh-grmip, which contains X integrals, is equivalent to the X linearly independent quantities where ftW-X.+ (''7')x-.f+(''2')x-.f + - for r = ], 2, ..., X-' the quantities %,, ,.,, y^^ are periodic functions of z, but they are not necessarily uniform : f denotes — , and fr-l\_ (r-1) ! _ Note. By taking ;:^„ = q>~"0„, for m = 1, ..., \, and wiiting the integrals become e. (^) =4>.+ ('■ ^ ^)>._,^ + r ^ ^) ^^^^ + . , . the functions i^ having the same character as the functions )(. 135. There is a theorem of the nature of a converse to the foregoing proposition, which is analogous to Fuchs's theorem proved in ^ 25 — 28. The theorem, which manifestly is important as regards the reducibility of a given equation, is as follows :— If an expression for a quantity u is given in the form w = ^" \4,n + ^„_i? + 4>„^,K' + ■ - + W-" + W-'l where & is a constant, all the functions cp-^, ..., 0„ are periodic in a>, and ^ denotes - , then u satisfies a homogeneous linear differential y Google 135.] CONSTRUCTION OF UNIFORM INTEORALS 421 equation of order n, the coeffidents of which are uniform periodic functions of z, Iiaviiiff the period a; -moreover, are integrals of the same equation and, taken together with u, they constitute a f undo/mental system for the equation of order n. The course of the proof is so similar bo the proof of the corre- sponding theorem as established in §§ 26 — 28 that it need not be set out here*. It can be divided into three sections ; in the first, it is proved that — , ..., ■ -^ satisfy such an equation, if u satisfies it ; in the second, it is proved that these must form a fiindamental system, for no homogeneous linear relation with non- evanescent coefficients can exist among them ; in the third, it is shewn that the linear equation, which has these quantities for its fundamental system, has uniform periodic functions of z with period w for its coefRcients. The details of the proof are left to the student. Mode of obtaining Integrals that are Uniform. 136. The further determination of the analytical expressions of the integrals, on the basis of the properties already established, is not possible in the general case. Thus the functions )(i, ..., %a> occurring in the sub-group specially considered in § 134, are periodic functions of the second kind with a multiplier ^. If we take new functions ■^i{2), ..., "^/.(f), such that these new functions are periodic of the first kind. But further properties of the functions must be given if there is to be any further determination of their form. When we limit ourselves to the consideration of those equations whose integrals are uniform functions, (criteria are determined * Some of the analysis of g 133 ig useful in establishinf; the theorem. y Google 422 EQUATIONS HAVING [136. independently by considering the integrals in the vicinity of the singularities), some further progress can be made ; bub, of course, the assumption that the integrals are of this character must be justified by appro|)riate limitations upon the forms of the coeffi- cients Pi, ..., pm in the original dift'erenbial equation. In such cases, every quantity such as -^^C-^) i^ a uniform simply-periodie function of the first kind; it can therefore* be expressed in the form of a Laurent- Fourier series such as Such a, form of expression does not lead, however, towards the determination of the criteria for securing such a result or any other result of a corresponding kind for any other assumption. In particular examples, we adopt a different method of practical procedure. In order to determine some of the functional properties of the integrals, it frequently is expedient to change the variable so that, if possible, the transformed equation belongs to one or other of the classes of equations considered in preceding chapters. Thus if the coefficients^!, ...,pm, which are uniform periodic then, introducing a new variable (, where we obtain a linear equation, the coefficients of which are rational functions of t. Some characteristic properties of the integrals of the equation in the latter form can be obtained by earlier processes ; it may even be possible to determine the fundamental system of integrals. The preceding transformation is, however, not the only one that can be used with advantage ; and it often happens that the special form of a particular equation suggests a special transformation which is effective. In pai'ticular, if the coeffi- cients in the equation are alternately odd and even functions, ' T. F,, % 112, y Google SJMPLY-PERIODIC COEFFICIENTS such that pi,pi,Pi, -.. are odd, and p2,pi,pe, ■■■ are even, then wc may take as a new independent variable r it is easy to prove that the transformed equation has uniform functions of ( for its coeffi- cients. Also, some indication is occasionally given as to a choice between these two transformations ; for example, if an irreducible pole of the original equation is a = 0, we should choose ( = sin — aa the transformation, and consider the integrals in the vicinity of t = 0; whereas the other would be chosen, if an irreducible pole of the equation is s = ^oi. Another transformation, that sometimes can prove effective, is any uniform function of 3, periodic in o>, can be expressed as a uniform function of t ; and the differential equation is transformed into one which has uniform functions of t for its coefficients. Ex. 1. Couaider the equation J'+2.:^cot.+ (6 + .oot..)„.0, where a, b, c are constants. Writing we have the equation whei'e and the oquatio + (/3+T-cot2s)», = O, The indicial equation for t—0 is p(p-l) + y=0. If v=Saj''*'' y Google 424 EXAMPLES [136. satisfies the equation, we have n2+2n(p--2) + 4--0-3p '• if+M^ir^) *■"'■ The form of a„, in terms of o^-a, shews that the series for y converges for values of ! ^ 1. If the two roots of the indicia! equation arc p^ and p^, and f{i,Pi),f{t, Pa) ho the two values of y, the primitive of the original equation is ,>— in-. |J/(siii 4 p,)+il/(«n ,, a)}. Ex. 2. Consider the equation we find the transformed equation for i to be which is Legendre's equation and so its primitive is Ex. 3. Obtain int^rals of the equations -j-5 +-J- cot!--wcoseo^3=0; d^ dz d^ die a a a —ft. dz^ da ~ ' (i) (ii); (-)S-(s-nr^-OSHovk-.-i.^)^- Ex. 4. One integral, /(^), of the equation 4(i!-,i„.)5 + !.(3.m. + 2c„..^«)* + (5-3co..-,m.),.0 the relation find the general solution. (Math. Tripos, Part it, 1896.) Ex. S. Shew that the equation has an integral where S[ has an appropriate constant value; and obtain the primitive. (M. Elliott.) y Google 136.] liapounoff's investigation Ex. 6. Obtain an integral of tlie equation where A is a constant, in the forin ^_sin(?-gi) sin(a-33) ^(cots^+cots,) where 2, and e^ are appropriate determinate constants ; and obtain the primitive. (M. Elliott) E.V. 7. Integrate the equation where w is an mt«ger, and A is a constant. (M. Elliott.) 137. A somewhat different form of the theory is developed by Liapounoff*, whose investigation deals with a more general equation, given by where /i is a parameter, and p (s) is a uniform periodic function of period €0. Let f(z) and ^ (e) he two integrals of the differential equation, respectively determined by the initial conditions /(0).1| *(0).01 /'(o)=or f(o)=ir Then we have relations of the form and the equation for determining the multipliers is (n-n)(n-8)-^7 = 0, that is, a= - (a + 5) fi + 1 = 0, as in § 127, Ex. 1. Clearly, we have * Comptes Hendus, t. cxxm (lS9e), pp. 1248—1252; ih., t. cxSTiii (1899), pp. 910—913, 1085-1083. y Google 426 liapounoff's [137. BO that, if we write the equation is Writing ;"-24n + 1.0. = A+(A'-l)t, and assuming that A'—l does not vanish, we obtain two integrals in the form where Fj (s), F^ {z) are functions of z, periodic in to ; and thus the complete priniitive of the equation can be obtained. The actual expressions for J'i(s) and F^{z) can fee constructed as in the preceding sections ; and the value of p depends upon that of A . When ^ = 0, the primitive of the original equation is shewing that the equation for dotermining the multipliers is (li- 1)^ = 0; and then A = \. Hence, when /a is not zero, and when A is expanded in powers of /*, it is inferred that A is oi the form ^ = 1 — fi,Ai + f>?A^ — iJ?A,^ + .... When .^1, A3, ^3, ... are known, the two values of il, which satisfy the equation n'^-24 11 + 1=0, can be regarded as known, and the primitive of the dift'erential equation can be obtained. For the purpose of obtaining the value of A, which is 4 = i(/W + f(»)l. where the integrals f{z) and <^ {s) are deiined by the initial conditions, we assume both f{z) and {z) expanded in powers of f{z) = Wo + fiM-i + ij?u.i + . . . ; then, in order that it may satisfy the equation yGoosle 137.] EQUATIONS we have and so on , From the first, we have Ik = a, , + *.«; from the second, we have from the third, we have = ai + b,z — j rfy I u„(a:)p{x)da:; ve have = C!a + 632 - 1 dT/j «i (x) p («) da: ; and so on. Now /(0)=1, /'(0) = 0; accordingly, a^ + /j.a, + f>?-a.i + . . . = 1, ha + iih^ + f.% + ... =0. Taking account of the fact that /j, is parametric, we have «„ = !, a, = for s = l, "2, ..., ?i, = for s-0, 1,2, , and thus we have M„=l, Wi = -j (^^f p(a:)da:, and so on. The value oif(z) is given by /(s) = 1 + ^Mi + /t'Ms + . . , . Similarly for (2), which is determined by the conditions <^(0) = 0, .^'{0) = 1; its value is given by <j>{s) = e + fiVi + /iH'a + . . . , y Google 428 liapounoff's [IS?, where Vi = -j dyj xp{a;)dx, V, = -\yy\yia:)v,{x)dx. and so on. We require the quantities /(w) and '^'(oi): let thorn be denoted by where [/j = — I dy\ p{x) dx, T, = — I cnp{(£) dx, and so for the others. Substituting the value of ^ in the form -4 = 1 -/tjl, + /iMi. — ..., we have = 1 dy\ p{x)dx+ i xp(x)dx = j %j ^(«)'^ic + j yp(y)dy. But ^ 1^ f^^i? (a') (^a^l = f V (*) -^^ + yp iy) ; integrating between the limits and a, we have Next, we have iA, = -11.-r, To transform these definite integrals, we write j'"j,W<fa-PW, P(»)=n. y Google 137.] METHOD 429 so that uAi!) = -t'dtf'p(6)de = -r P{t}dt, v,(a:) = -!'dtj'Sii{e)de ~-rtP{t)dt+rdtl'p{S)de = {'dti' {F{lf)-P(t)]ie. We have J- 1"! {s)j^p(s) d!/\ - «. (y)p to) -P'(s); therefore and thus the first integi-al in tlie expression for 2A^ is equal to j'dyj'{F{g}-F(a:)]P(^)d^. Similarly, we have therefore = -nj'\P(!,)% + n|^"<!yJ'p(a.)(i» + j''yP-(y)*-JJ<ij/'P&)PW<i«:. The first and third terms on the right-hand side together are --JJ(n-P(y)|P(y)t,<iy = -/J.iy/V-P(y)lP(y)<fc. y Google 430 liapoukoff's equations [137. ao that I" V, (:v)p (.^) <^ = - /^" d^j' [ii'P (y)} \P iy) - P (.^)} dx, which gives a traasformation of the second integral in ^A^. Combining the results of transformation for the two integrals in 2-^2 , we have HA, = 1^°' d>j r {il-P{y) + P {x)} {P (y) - P (^)i dx. Similarly, it may be proved that the value of ^A^ is j'Ji/J%/^(n-PW + PW!|PW-P(y)HP(s)-PWi<fc, and so on i so that the value of A, and therefore the value of P, is known. The investigation is continued by Liapounoff, especially for the purpose of discussing the values of fi. which satisfy the equation ^=-1 = 0; aad the results appear to be of importance in the discussion of the stability of motion. The reader is referred to the notes by Liapounoff already cited (p. 425, note) ; other references to more detailed investigations are there given. Esc. 1. Establish by induction, or otherwise, the general law for the coefficients A, viz. 2.t„= <h!,\ rf%... I @rf«„, where ©.(ii-P(i,)+P(«.,)){fW--PW]fi'W--P(».))... If ('.-J-i" (».)>■ Ex. 2. Shew that, if the peiiodic function p(s;) always is positive, then all the coefficients A are positive; and prove that Hence shew that, when p (x) is positive and satisfies the inequality y Google 137.] EQUATION OF THE ELLIPTIC CYLINDER 431 Ex. 3. Prove that, if the periodic function p {m) be real and odd, so that the series for A contains only even powers of /t, then A^= 4 I °'(fej r'dx^ fds;^ i"' [P^- P^f {P.^~ P^f dx^, and MO on, where P,. denotes i'i^r)- Prove also that, if tho constant a being determined so that I PciK=0, and if "" P''dx:^i, 'I? then ji^<I. Ex. 4. Discnss the values of )i which are roots of the equation (All these results are due to Liapounoff.) Discussion of the Equation of the Elliptic Gylindbe. 138. One of the most important equations of the ciass, which has been considered in § 137, is the equation commonly called the equation of the elliptic cylinder; it is of frequent occurrence in mathematical physics and astronomical dynamics. It forms the subject of many investigations* It is known (g 55) to be a transformation of the limiting form of an equation of Fuchsian type. Moreover, it has already (| 127, Ex. 1) been partially discussed in connection with another equation and for another purpose. In this place, it will be brought into relation with the preceding general theory. Let new independent variables u ov v h& introduced, such that * Heine, Handbuch der Kugelfanctianen, 1. 1, pp. 404- — 415; Lindeniann, Math. J)iji.,t. XXI! (1883), pp. 117—133; Tisaerand, Mecanique Cilcste,t. in, oh. i, at the end of which other referenoes are given. y Google 432 EQUATION OF THE [138. The equation becomes .d^w , ,, „ , dw , , ^ , "* *^ ~ "^ rf^? "^ *^^ ^ ^""^ rf^ + ^^"^ ~ ' + ^'^''^ "^ = "^' when u is the independent variable ; and it becomes , dhi) , ,, „ , dw , , ^ , when V is the independent variable. Accordingly, if is an integral of the equation, another integral is provided by »=/(., -c). The indicial equation for m = is p{/'-^) = 0; if w = ta^vP+i' be the integral, the scale of relation between the coefficients ap is (p + p)(p + p-^)S=KP + p-l)°-i(a-c))ap-i-^cap_a, with the relations «o= 1, When p = 0, let fflp = ^ {p, c) ; when p = ^, let aj, = ?y (^, c). Then two integrals of the equation are X,- i u^e{p, c), a^i= i ?!P+sa(^, c), M'ith the convention t)(0, c) = l=Sy(0, c). It is clear that, when z is Jtt, so that m is 0, ..=,, J-=o, ' ' dz -1. r, as the equation in w i s satisfied by » „ and dx, dx C y Google 138.] ELLIFHC CYLINDER 433 But (^M = - 2 Bin s COS ads = - 2 [u (1 - u)\idz, so that ' ds " dz When 2 = ^7r, the left-hand side is equal to 1 : hence and therefore, for all values of z, we have dxa dxi __ ^ dz " dz Two other integrals of the equation are given by they are such that, when z is 0, and therefore v is 0, ■^° dz ^ dz and, for all values of s, dy-i dtig ■^^ dz ■^ ds Now when z is real, both m and v are real and lie between and 1 ; and, in particular, when 3 = ^7r, then « = •«= |. For such values, x„, x,, y^, y-,, coexist ; and so we have relations of the form where a, (S, -/, S ai-e constants. Hence y. (i) = ™. G) + /3a>, (i), -</.'(« = «; (i) + Hi'l (J), where and so for the others. Hence « = -j.(i)<a)-y.'(i)«.(i). /3= y.^^/ffi + y.' (»■».&)■ Similarly 7 = -</.(!)<(» -y.'(i)''.(i)l S= J(.tt)»=.'(i)+!/.'(l)«.(J)l' F. IV. y Google 434 EQUATION Of THE [138. and it is easy to verify that Moreover, we have ■*» = %!. - /3i/il ^1 = - 7^0 + «yJ ■ 139. The integrals x^ and a;, are valid in the domain of w = ; tho integrals y„ and y^ are valid in the domain of v = 0, that is, of u= 1. Lindemann* proceeds, as follows, to obtain uniform inte- grals valid over the whole of the finite part of the plane. After a small closed circuit of u round its origin, x^ returns to its initial value and «, changes its sign ; hence y^ becomes aic„— /3«], and j/i becomes yxn — hxt. After a small closed circuit of u round the point 1, the integral i/o returns to its initial value and y, changes its sign. Consider a quantity ij, where jj = Ay^ + %l^ as a function of u. It remains unchanged when u moves round the point 1. Its two values in the vicinity of it = are {A-x' + £7=) x^ + {A^ + £8=) x^ + 2 {Ai0 + B7S) x,x^, {Aa^ + Brf) x," + {A^^ + BS') x/ - 2 (Aa^ + SyS) x,x„ which are the same if Aa^ + By& = l}: henee the function is uniform in the vicinity of u = if this condition is satisfied, that is, the function is uniform over the whole plane. The condition is satisfied if we take and then n = tt^y," - ySyo". Moreover, in the region of existence common to y^, y,, x„. a;,, we '^^yi' — 7^3/11° = 0Sxi^ — ayxa\ Hence defining the function 71 in the domain of 3= by its value in terms of y^ and j/i , and defining it in the domain of 2 = 1 by its value in terms oi x^ and a-'i, we have a function ^ = F{u) = F(>Ms's)=^ (i), • Math. Ann., t. xxii (1883), pp. 117—123. y Google 139.] ELLIPTIC CYLINDER 435 say, which is regular in the vicinity of m = 0, regular in the vicinity of w = 1, and therefore is regular over the whole iinite part of the z-plane. Now let F,-y,(a/3)t + s.(7S)ll 7.-!,,(.«)l-S.(,S)tf' then = -2(=i/37S)l. Also and therefore l'«t-+5^.?^" = <E>'(^)- ds dz Hence Y, dz ^ ^{z) <E>(2) ' 1 dY,_,^'(z) (a^78)'. and therefore where These integrals of the original differential equation are valid over the whole of the finite part of the plane. Accordingly, we may take two integrals M {-- (?,(2) = {a>(2)j4e' -'*w as integrals, which are valid over the plane and have z—'X for their sole essential singularity. We now proceed to shew that they are uniform over the plane. Substituting in the original differential equation, we have {a + c cos Is) <^-' - 1^'-' + ^4><l>" + M= - ; y Google 436 lindemann's [139. so that, as if in general is not zero, any root of 'I' = is a simple root. Let k denote such a root : then Now let z describe a simple closed contour, including k and no other root of $ = 0, and passing through no root of "5 = 0. Then, at the end of the contour, |*^(2))S has changed its sign. As for the exponential factors in G{z) and 0,(z), they are multiplied by respectively, the integral being taken round the contour, that is, they are multiplied by ^m!'}-^, {k) that is, by —1. Thus G{z) and G^(s) are unaffected by the contour; they are therefore uniform in the vicinity. Moreover, in the immediate vicinity of k, we have ^(0) = (z-k}^'{k) + ..., so that GA^)={<^'(k)]^(^-k)e-^^'-''^Q(z-k), so that A: is a simple root for one of the integrals and it is not a root for the other. Similarly, in the vicinity of any other root of ^ = ; hence G and G, are uniform over the whole plane. Now take any path from z to s + tt, for tt is the period for the original equation. We have where F is uniform ; hence i,(,+ ^)—H^), {*{^ + ^))4 = (-l)'jO(2)jS, where r is or 1, depending upon the path from to tt. The effect upon the exponential factor of G (e) is to multiply it by yGoosle 139.] METHOD 437 Wo know that ^{s) is regular over the whole plane, that it is periodic in ir, and that it has only simple roots ; hence, taking a path between z and a + tt, that nowhei-e is near a root, we can 1 valid everywhere in the range of integration. Then and, consequently, if then Similarly Hence G and G^ are the two periodic functions of the second kind, which are integrals of the original equation* ; and they have been proved to be uniform functions, regular everywhere in the iinite part of the plane. Es. Shew that the equation has two particular mtegrals the product of which is a, single-vahied tnius- cendental function. F{z) ; acd shew that the integraLs are "'-'"•""'■"-■[-"/ (.(i^'i^w ]- where C is a determinate constant. In what circumatatices are these two particular integrals coincident ^ (Math. Tripos, Part ii, 1898.) liO. The multipliers /m and — are thus the roots of the equation D.^- in + 1=0, ' This ineluaion of Lindemann'a apeeial result within the general theory is dua to Stieltjfls, AetT. Naehr., t cii (1884), pp. 147, 148. y Google 438 INVARIANT OP THE EQUATION [140. where the invariant / of the period o> is Another expression for this invariant, consequently leading to another mode of obtaining these multipliers, has already been given in Ex. 1, § 127. Both processes are dependent upon the determination of simple special solutions of the original differential equation. Another method of proceeding is as follows. Let so that so that, if (?(£)=e*''©(3), then, as 6(z) is a uniform function of a, regular over the whole plane, (s) is a uniform periodic function of the first kind, regular over the whole plane ; and ir is the period. Hence we have and therefore Now in the vicinity of a- = 0, the integral y^ is even and yi is odd ; hence G(z) contains both odd and even parts. The form of the differential equation shews that, if /(z) is an integral, then f{—z} also is an integral ; hence, as (a) exists over the finite part of the plane, G(—z) also is an integral. Henco, taking where a is an arbitrary constant, it follows that Il{z) is an integral of the original equation, which exists for all finite values of z. Substituting in the differential equation, and noting that cos 2s cos [(2m + h)z + a\ = ^ cos {(2n -2 + h)s + ix} + ^cos [(2« + 2 + h)s + a.\, we have "s" i{a - (2n + hy\ K„ + ^v (>c„^, + Kn+,)'] cos {(2n + h)s + u}^0. yGoosle 140.] OF THE ELLIPTIC CYLINDEK 439 as an equation which must be identically satisfied ; hence {a - (2k + hf] «„ + Jc (k„_, + K„+0 = 0, for all values of n from -co to + a^ . The mode of dealing with this infinite set of equations by means of infinite determinants has been indicated in a preceding chapter, and much of the analysis of the first example in § 126 is directly applicable here : so we shall not further discuss this mode of obtaining h and the ratios of the coefficients «. There is, however, another method of obtaining these quantities: it is due to Lindstedt* and is specially adapted to the differential equation under consideration, for purposes of approximation when c is con- veniently small. Writing «,. = 2 (2ii + hy- - 2«, ?' C 1- c <Xn a„a„+, a„+,a„+<j 1- 1- I- Owing to the form of — for increasing values of r, it is easy to prove that this infinite continued fraction converges, for all values of n. We therefore have Similarly K„ 1 - 1 - 1 - inf c t= c'^ «_„ o_„a_„_i a_„_ia_n-. '■ ... ad inf.. ,, 1- 1- 1- 1. dn I'Acad. Si Pitersbourg, t. xxx 1 (18B3), No. 4. y Google 440 lindstedt's [140, which is a converging continued fraction ; and, in particular, c c" d" /Co 1 — 1 — 1 - ■ ■ ■ But, from the fundamental difference -equation, therefore g2 g3 (.2 p2 p! C^ 1 _ °^»"i °i^ C2W3 , °ut(-i 0.^1 a_j a-;0:-; 1 - 1-- l-"""^ 1- 1- 1- ■■■' a transcendental equation to determine h, which of course is equivalent to the corresponding equation arising out of the vanishing of the infinite determinant 1) (p). Denoting the first continued fraction by - and the second by ^ , so that these values may be regarded as coiivergents of infinite order, we easily find r=a «=r+a (-S+2 OrOr+i a,a^i Ojat+i J o = 1 - i -^ + i i -^ ''' ■ -si i — ' ^^L.+ ...; r=lj=r+2 (=8+2 ar«r+I aa«s+i OfOi+i the values of ^' and q' are derivable from the expressions in p and q respectively, by changing a^ into «_„ (for all values of /t) wherever a^ occurs. The equation manifestly lends itself easily to successive ap- proximations. Thus, if we neglect C and higher powers, we have which, to this order of approximation, gives The calculation of the coefficients can similarly be effected. y Google 140.] METHOD Pi-ove that, up to sixth powers of c inclusive, < -i , 1 <^ 1024c[2(l-a)3(4-a) 105a'-H55a'+3815w=-4705aHl653(t- (In astronomical applications, a it pared with a.) Ex. 2. Taking k^=\^ and writing prove that, up to e^ iuclusive, 1(1-.)' (4- mally not ai ger, and c is small com- (Poincar^, Tiaserand.) U+? 10243(l+j)9(2 + s')(l-g)r ^\\-q 1024j(I-9)'(2-y)(l+2Jj (|c)=f coa(Z+fe) , _oo8(£-4^)_l ■^ 2! l(l + j)(2+j)'^(l-y)(2-3)J (i«:^f c oa(g+6. ) , eoa{Z-_6^ 3! \(l + 5)(2+3)(3 + j) whore y^^ "(l-j)(2-S)(3-y)J' (Poincar^, Tisserand.) Ex. 3. In the investigation of § 138, the quantity if is supposed to be different from zero. When M is zero, the integrals Q{z) and Q^if) aro effectively the same ; and neither of them is uniform, so that the remainder of the investigation does not apply. Discuss the case when j¥=0. (Heine.) Equations i [ Uniform Doubly- periodic Coefficients, 141. We proceed now to the consideration of linear equations, the coefficients in which are uniform doubly-periodic functions of the independent variable. Let the equation be = 0, where p^, ..., p,„ are uniform functions of z, which have no essential singularity in the finite part of the plane and are doubly- periodic in periods to and w', such that the ratio of oi' to w y Google 442 DOUBLY-PERIODIC [141. is not purely real. A fundamental system of integrals exists in the domain of any finite value of z, and may be denoted by /.W. /.(') /-«, which accordingly are linearly independent of one another. The differential equation is unaltered when 3 + lo is wiitteii for 2 : hence /,(^+«), /,(^+„) /„<,+») are integrals of the equation and, as in § 128, they constitute a fundamental system of integrals. Similarly, as the differential equation is unaltered when a + to' is written for 3, /.(«+»'), /.(^+»') /■(^+»') constitute a fundamental system of integrals. Choosing therefore a region common to the domains of these three fundamental systems (a choice that always can be made because the singu- larities of the integrals are isolated points, finite in number within any limited portion of the plane), we have relations of the form /, (« + «) = o„/,(^) + ...+o„/„W| /. (^ + " ) - «»l/l («) + ...+ 0,.»/,(2) and /. (^ + .-)_4,. /.(.)+. .. + i,./„W /,. (2 + »■) . J,,./, W + . . . + (.„/. W ) valid within the region chosen. The coefficients a are constant, and their determinant is not zero ; the coefficients b also are constant, and their determinant also is not zero. The two sets of relations may be represented in the form /(^ + »)-s/w, /(^ + »)=.S7W, where S and S' denote the linesir substitutions in the relations. The coefficients in the two substitutions are not entirely independent of one another. We manifestly have /,((^ + .) + »-l-/,l(^ + »■) + -!, for all values of r. The symbolic expression of this property is /((. + ») + ..■) = S'f(z + „ ) = S'iv w, /!(. + «■) + « I = s/(« + »')-ss7W, y Google 141.] COEFFICIENTS 443 80 that or the linear subsfcitutions are interchangeable. The explicit expression of relations between the constants is obtainable from the equation br:A (^ + «) + &,./. (^ + a>) + . . . + 1™/,„ (^ + o>) = a.,/, (z + 0.') +«„/, (^ + »')+... + a™/« (^ + o>'), by substituting for the functions / (3 + w) in the left-hand side and the functions /(z + o>') in the right-hand side. The result must be an identity, for otherwise there would be a linear relation between the members of the fundamental system f,(2;), ...,/„(«); hence, comparing the coefficients of fg (z) on the two sides after substitution, we have aay. This holds for the m" equations that arise from the values r, s= 1, ...,m. Of the m' equations, only m^ — m are independent of one another, a statement the verification of which (alike in genera!, and for the special values m = 2, m = 3) is left to the reader : it can also he inferred from some equations which will be obtained immediately. The number of the relations is less important, than their existence and their form, for the establish- ment of Picard's theorem relating to integrals with the character- istic property of doubly-periodic functions of the second kind. Consider a linear combination of the members of the funda- mental system in the form F(^)-\A(z) + KMz) + ...+-K,.f^{^), where X, will be taken as equal to unity when it is not bound to be zero; and let the constants X^, ..., X^ be chosen so that, if possible, the relation is satisfied, 6 being some constant. To this end, we must have \,0 = \a^^ -l-Xjaai +X3asi + ... 4- X,„a,„| , X^0^ n + X3%,„ - fX,„am„„ y Google 141 and therefore FUNDAMENTAL EQUATIONS [141. the equation satisfied by 0. As in the case of the single period in § 128, it may be proved that this equation is independent of the original choice of the fundamental system of integrals /i(z), ..., fmi^)- The coefficients of the various powers of 6 are therefore invariants, and the equa- tion is called the fundamental equation for the period o). Now let = \^1W -\A^ + \shsm I- \»b,„rn- Multiply the earlier equations, which define the quantities X and lead to the equation fL(B) = 0, by b^r, b^, ■■■, &«»■ respectively, and add: then Ofir = \ (Oii fcir + «12 tsr + ■ ■ ■ + tim f>mr) + 'K (<*!! 6,T + «32 6sr + - ■ ■ + Ossn hmr) + Ki (Clnil &ir + ChaAr + ■■■ + amm^mr) = \i (b„ a,^ + bi3 a^ + --.+hm O-mf) + Ki (6r«l«]r + h^a^y + . . . + imma,nr) This holds for all values of r ; and thus we have -nr" = '^ifim + ^ «»ni + ^ ttsni + •■■ + "3'" '^mitf y Google 14.1.] When the uniquely FOR THE PERIODS 445 ; compared with the earlier equations, we have for all values of r; and therefore the same values of X^, ..,, X„, that enable the equations connected with the period fu to be :ad to the equations X„^=Xi6im + X2ijm + ?^-3&3m+-.-+X^&mm. Hence J?(2 + «)-X,/,(« + »') + X.,/.(^+«')+--- + V/,.(^ + «') - 9' (A,/, w + x,/, (^) + . . . + \,u Wl on using the prece satisfies the equati Q.'{e') = ing equations. Moreover, th n b,^-e\ b^ , ..., b,„, b,i , h^-ff, ..., i,„a IS multiplier 6 = 0. h«. , b^ i™„-^' This equation, like il {B) — 0, is independent of the initial choice of the fundamental system of integrals /i (z), .,., /m{^), the proof teing similar to that in | 128. The coefficients of the various pow&rs of 6' are therefore invariants; and the equation is called the fmidamental equation for the period «'. The term independent of ^ in II {$), and the term independent of 6' in il'{ff), can be obtained simply. Let A (a) denote the determinant d-'A ■< "/■ d'^A da™-' ' ds™-^ dn^-' AK df, dz f. dU dn y Google 446 FUNDAMENTAL EQUATIONS [141. then, as in § 9, we have -.-. — - . Hence I p, Ix) dx so that A (z + ») _ f'*"r, (•)<!• aw "~" ' and similarly where we manifestly may assume that the path of integration does not approach infinitesimally near the singularities of jj,. Now 'pi is a uniform doubly- periodic function with no essentia! singularity in the finite part of the plane; if, therefore, a^, ,.,, a„ denote its irreducible poles, and if f (s) denote the usual Weier- strassian function in the same periods w and a' as Pi, we have* with the condition Now j'*'p, W d^=C^+SA, log ^-(t " v""- + l^ii,{ir(2 + „-«,)-f(^-«,)l -Co.+ I ^,|i7r + , ,0 + 2, («-»,)!+ I 2,i(, = Co) - 2, S 4,0, + 2, i a, - D, ' r. 7''., g 129. y Google 141,] OF THE PERIODS 447 say ; and similarly say. But, substituting in A (s + w) the expressions for f^ (z + a), ..., /m (2 + w) and their derivatives in terms of y, (jr), . , . , f^^ (z) and their derivatives, we have which is the non-vanishing constant term in il (0). Thus and similarly il'((9') = e»' + K-ir^'™ In particular, when p-^ is zero, so that the differential equation has no term in -^r~ — r , we have I> = 0, D' — Q; and thon il(6) = \ + ...+ (- 1)"-^, il'(^')= 1 + ...+(- 1)-"!?"". Integrals which are Doitbly-periodic Functions of the Second Kind. 142. Let ^ be a root of the equation SI {&) = 0. Then quanti- ties X,, ...,Xm exist such that the equations leading to ii(^) = are satisfied; and a quantity 6' is obtained, when the values of Xu, ..., X,n are substituted in its expression. It thus follows that there is an integral F{z) of the differential equation such that F{z + a,)^6F{z), F{z + o>') = e'F(z), where $ and d' are constants. Such a function is called* doubly- periodic of the second kind : and therefore it follows that a linear differential equation, which has uniform doubly-petiodio functions for its coefficients, possesses an integral which is a doubly -periodic function of the second kind: a result first given by Picard. * T. F., g lae. y Google 448 picard's [142. When d is si simple root of the equation fi (S) = 0, then Xu, .-.,X™ are uniquely determinate: and 6' is uniquely determ- inate. When S is a multiple root of its equation, quantities "Kt, ..., Xm exist satisfying the associated equations but they are not uniquely determinate : and assigned values of X^, -.-, X™ determine ff. Similarly for ^ as a root of the equation li' (6") — 0. Combining these results, we have the theorem* i A linear differential equation, having doubly-periodic functions for its coefficients, possesses at least as many integrals which are doubly-periodic functions of the second kind as either of the equa- tions li (0) = 0, H' (6') = has distinct roots. By using the elementaiy divisors of fl {$) = 0, we can obtain a more exact estimate of the number of integrals which are periodic functions of the second kind, associated with a multiple root. Let 6, be a root of il{6) = of multiplicity Xi, and let ii^ be the number of different elementary divisors of H (6) which are powers of ^ — ^j, so that the minors of il (0) of order m, are the first in successively increasing order which do not vanish simul- taneously when = $,. Then (§ 133) the number of integrals, which satisfy the equation IS precisely equal to «i. * These equfttions appear to have been oonsidered first by Pioard in eenoral; see Comptes Remlris, t. sc (1880), pp. 12&-131, 2<)3— 295; Creile, t. xc (1880), pp. 281—303. Their properties were farther developed by Floquet, Gomptes Bendvs, t. xcviii (1634), pp. 82—85, A-rm. de V^b. Norm. Sup., 3™ Sir., t. I (1884), pp. 181—238, \Thich should be consulted iu oouueotioii with many of the following investigations. A proof of Picard's tbeorem, different from that in the test, is given by Barnes, Messenger of Malltematici, t. sxvii (1897), pp. 16, 17. Investigations of a difierent kind, leading to equations the primitives of which are espreasible in terms of doubly-periodic functions, are curried out in Halphen'a memoir "Sur la reduction des equations <liff4rentiel!es lin^aires aux formes int^grables," Mim. des Sav. Strang., t. xsvni (1883), No. 1, 301 pp. ; particularly, chapters ii and ii. The most important equation of the type under consideration is the general form of Lainfi'a equation. It had heen considered by Hermite, previous to Picard's investigations; and it has formed the subject of many inemoirH, references to some of which will be found in my Theory of Fumtiois, ^% 1S7— 141. y Google 14;2.] THEOHEW 449 Moreover, in that case, tii of the equations in § 141 for determining the quantities X are dependent upon the remaining m — Ui. Let the last m — Ui be a set of independent equations, determining Xn^+,, ..., \m in terms of \, Xg, ..., X„^ ; and suppose that the expressions are Xj = /;,iXi + k^\! + kgs\+ ... + km,'K„ for s = «, + !, n, + % ..., m. Then F (3) = \A (2) + X./. (2) + , . . + X„./,„ (s) = \(fi is) + ^^9^ (^) + . . . + X„, 3,,, {2), where <,,(»)=/,w+_J^^t/.(.), for r = l, 2, ..., n,; and each of the functions g^, ...,^,i, is snch that j,(2+«.)-e).s,(.). Ah regards the possible multiplier 6-^ for the other period, we have 6^ = Xii„ + X2&„i + X,6s, + . . . + X,„J™i = x,A + M, + ...+x„,s„,, say, where and the effect upon F {s) of the increase of argument by the period <o' is given by Now 6i is not zero, for it is a root of O' {&) = which has no zero root ; and therefore not all the quantities B^, B^, ..., Bn, can vanish. Let Bi, B^, ..., B, be those which do not vanish ; then we have X.sf, (s + 0.) + X^g^ (^ + w) + . . . + X„,^„, {z + <o) = (XiS, -H X,B, + ... + X^B,) [\g^ {e) + \^g^ (s) + ... + X„,^„, (s)}. As some one of the quantities Xj, X^, ..., X^, is not zero (for, thus far, all these quantities are arbitrary), we shall take X,= l. In order that this equation may hold, we assign definite values to Xj, ..., A,; we write B, + \B,+ ... + X,B, = 0,', g, (^) + X.i'. (2) + . . . + X,5-, (z) = G (^), F. IV. 29 yGoosle 450 DOUBLY-PEllIODIO INTEGRALS [142. and then, as Xg+i, ..., >.«, t^n remain arbitrary, wo have forr = s + l, ...,K,. Moreover, on account of the composition of Q (z), we have and we had Accordingly, the nmnber of integrals, which are doubly -penodic functions of the second kind and are associated with the multiple root 0, of the fundamental equation ii{^) = 0, is where % is the mimber vf elemental y divisors of ii(^) which are powers of — 0,, "nd s n the number oj quantities which do not vanish, so that < s < Ki- 143. We now can indicate the total number of integrals, which are doubly-periodic functions of the second kind. Let 6, be a root of multiplicity X, of fi {6) = 0, and let it give rise to n^ elementary divisors of fl (8) which are powers of ^ — ^j ; and let s, be the number of quantities in the preceding investigation which do not vanish, so that <5i <«] €X,. Let 0^, 03, ... be other multiple roots; and let X^, n^, s^; \^, jt,, Sj ; . . . be the numbers for them, corresponding to X, , n, , s^ for 0^ ; so that X, + \ + \+... = m. Then the number of integi'als, which are doubly-periodic functions of the second kind, is 2 (1+M,,_s,.). yGoosle 143.] OF THE SECOND KIND 451 In particular, if the roots of Cl(0) = O be all distinct from one another, a fundamental system can be composed of m integrals, each of which is a doubly-periodic function of the second kind; the constant multipliers are the m roots of il (0) = 0, and the corresponding quantities 0' derived from them, these quantities 0' themselves satisfying the equation O.' (ff) = (I. Moreover, the relation between the equations satisfied by and Xi, ..., X™, and the equations satisfied by 0' and X,, ..., X™, is reciprocal ; for each set can be constructed from the other as in 1 141. Hence, if either of the equations £1 (f) = and li' (0') = has all its roots distinct from one another, there is no necessity to take account of possible multiplicity of the roots of the other, so far as the present purpose is concerned : the implication merely is that one of the two multipliers has the same value for several of the integrals. Further, if 9 and 0' are two associated multipliers, each of them arising as repeated roots of their respective equations, we shall suppose, for the same reason as in the preceding case, that the construction of the doubly-periodic functions of the second kind is initially associated with that one of the two equations which has the repeated root in the smaller multiplicity. Multiple Roots of the Fundamental Equations and Associated Interralh, 144. Wo have now to consider the form of the integrals associated with a multiple root of II (8) = 0, the fundamental equation for the period w ; and we assume that the correspond- ing root of li'(^') = is also multiple, to at least as great an order of multiplicity. Denoting this root by 0, and the corre- sponding root of il' (6') = by 0', we know that there certainly is one integral, which is doubly-periodic of the second kind and has multipliers and 0': let it be denoted by .^, so that <t>, {z + m) = 04,, (2), 0, {2 ^ m') - e'<i., {z). Considering the integrals first in relation to the period m, we know (§ 134) that the number of them associated with the multiple root is equal to the order of multiplicity of 8: and 39—3 yGoosle 452 INTEGBAL8 ASSOCIATED [144. further, that this group of integrals is linearly equivalent to sub-groups of integrals of the form Wj = 01 + 330, + 3£"0a + S^(j>„ the aggregate number of integrals in the various sub-groups is equal to the order of the multiplicity of 0, and each of the functions is such that ^(^ + »)-«+ (4 In these integrals, 0^ can have any added constant multiple of 01 ; also 05 can have any linear combination of constant multiples of 0s and 01 ; and so on. All the functions 0, so changed, still have the multiplier 8 for the period w. Now u, has the multiplier ff for the period m'. The simplest case arises when some other integral of the group, say %, also has this multiplier $' for the period a>' : for then all the intervening integrals have this multiplier for the period co'. What is neces- sary to secure this result is that, first, ,(., (» + »■) +(z + «,■) *, (, + «■) = »'!*, (^) + ^^, wi, that is, and therefore ■ ».(^ + »') ,., ».w *,(»+»■) + " *«■ Secondly, we must have .^, (2 + »') + 2 (« + «■) ^, (« + •>') + (^ + »•)• * (» + »') -9'l*.(.) + 2#,W + ^-*(A, which, in connection with the preceding equations, is satisfied if 05 (s + 0.') -i- 2«'0., (s + a.') -H m'^0, (2 -I- «.') = 0-<l>, (z), that is, if y Google 144.] WITH MULTIPLE ; ROC Similai-ly, we must have 4>,(z + (, ?!-. ,*.(» + ■ '-} + Sa" ,*.o M^ + u *.o and so on. •■) *W Let ^(s) denote the usual "Weieratrassian ^-function, with periods oi and w'; and let 17, V denote the increments of ^(z) for an increase of s hy the respective periods, so that we have 1701' — Jj'fii = + Stt*, the sign being the same as that of the real part of a>' -?- ita. Then, if a function u{s) be defined by the equation .(»+« ) = »W, «(^ + « ■)-«« + „'. .(.+„■)-*■« ^.(2) "W. that is, the function on the right-hand side is periodic in m'. Moreover, <J3^ and 0, have the same multiplier for q>, and u {z) is periodic in w; hence the function on the right-hand side is periodic in w also. It thus is a doubly-periodic function of the first kind ; denoting it by i/^.^, we have so that ^)i^2 is a doubly-periodic function of the second kind, with the same multipliers as 1^1, viz. 6 and &. Similariy, we have 3 that the function on the right-hand side manifestly is periodic 1 m' ; and it is periodic in co on account of the properties of m and y Google 454 INTEGRAI£ ASSOCIATED [144, the functions i^. Denoting this doubly- periodic function of the first kind by i/fj, we have And so on in succession. The group of integrals, in the case id, can be represented in the form where the functions ^i, ^^, ^s, Xi' ■■■ ^^^ doubly -periodic functions of the second kind with the multipliers 8 and 8', and 145. Returning now to the less simple case, when not more than one of the integrals associated with the corresponding multiple roots can be assumed to be doubly- periodic of the second kind, wc know that one integral certainly exists in the form of a doubly -periodic function of the seoond kind with the multipliers 6 and 8". Denoting it by </ii(s), we use it to replace some one of the integrals, say fi(s:), in the fundamental system, which then becomes *(A /.(A .... /„('}■ We have *(^ + »)=»*,(<:), The fundamental equation for the period to ia n(x)^\ e-x, , , ..., =0; and so f is a root of of multiplicity less by one than its multiplicity for li {x) = 0. y Google 145.] WITH MULTIPLE ROOTS Similarly, we have relations of the form /, (3 +<c') = d^ 4,, (z) + d.^f.A^) + --+d^ /™ (^X the multiplier ff being a root of D.{{x) = -d.^-x, d^, ..., d,„, 1 = 0, of multiplicity less by one than its multiplicity for Xi' (x) = 0. The coefficients cj and dij must satisfy conditions in order to have /.|{. + «') + <.|=/.l(^ + <.) + «'!. for all values of k : these conditions are dn(hi + 'ir3Cig + ... + drmCrm=^ Crldu + Cndis + ■■■ + Crmdms = K. say, with the limitations Cn = 0, d„ = d'; c.s = 0, d,, = 0, it s>l. Owing to the fact that ^ is a root of Oi(,k) = 0, quantities *3, ..,, Km exist such that ^Ks^Cas + CssKa-l- ... +C,„sK„,, (s = 8, ..., m), ^ = Cai + Cja/ira + ... +Cm2«m- Let '^ = dss + dsiKi+ ... +d„,^K^, iTf-d^+dirK3+ ■■■ + datrK„. {r = 3, ..., m); then Sa-p = dir (c-a + Csa K3 + ... + Cms «m) + '^3r (Csa +C33 Ka+-.- + Cm3 Km) (^2 Car + "^ Csr + . - . + daiH Cmr + Kj (das Cffl. + rfa3 «»+■■• +'^sm Cmr) + + «™ (<4.aCw + C^iaCar + ■ . . + (4™C™.) ; CjA + Car ffa + CirO-4 + . . ■ + C™.(7™ ; y Google MULTIPLE ROOTS AND [145. holding for all values of r. Comparing with the oarlier equations in c, we have 0-, - ?»/<:,., for all values of r ; and thus the second set of equations is a- = rf^ + (;^Ks + ... + <^ffl3«m, ^/(:,.= (4. + 4,«:a+ ... +d,„K„, (r = 3, ...,m). Eliminating the quantities rc, we have so that 0' is the value of a. Now consider the integral X. W -/. w + «./, (2) + . . . + «./.. (t). We have X. (» + »)-/■ (2 +«) + «./,(» + ») + .•.+«,./,.<« + ») = ".*« + "»(«), say, and » (' + »■) -/■ {•:+•>') + «./. (« + »•)+... + «./„ (^ + »•) = 6,*,W + «'X,«, say. When Hj and 62 vanish, ^^ is doubly-periodic of the second kind; but in the general case, a.^ and b^ are distinct from zero. The property x,l(^ + ») + .1-»|(^ + «') + »! leads to no relation between 0,3 and b^. If the multiplicity of (9 as a root of 12 (^) = and that of 6' as a root of H' (^) = be greater than 2, so that and ^' are multiple roots of ilj («) = 0, flj (x) = 0, we proceed as above. The newly obtained integral j;, is used to modify the fundamental system by replacing y^, say, so that the system consists of A. X.. /. /»■ y Google 145.] ASSOCIATED INTEGRALS 457 Then, in the same way as above, it is proved that an integral ;:^ exists such that Since »K^ + .) + »■) -»!(« + »') + »), we find, on aubstitation, so that we may take c^ — Xa^, di^Xb^, where \ is any parameter. This parameter may clearly be absorbed into j^, by taking Xa-^V and also into a, and h^ by division. Thus our integrals 0i, ')(_^, j^j are such that Xa (s + » ) = 050, + a^x-' + ^X3. <^,(^ + a,') = '5'0„ And so on, until a number of integrals is obtained equal to the leaser of the orders of multiplicity of 6 and 6'. Thus the next integral is x* (say), where X. (^ + « ) = a,0i + («-j + ^o.) X= + a=Xs + ^X- . X, (s + <o') = b,<f>, + {h + \h) X, + b,x, + B'x. . 146. From these descriptive forms, we can proceed one stage towards the construction of an analytical form of the integrals. For this purpose, we introduce (as in § 144) the functions where the doubtful sign is the same as that of the real part of w' -^ i<it ; we have Then the various integrals can be expressed as non-homogeneous polynomials in i( (s) and v (z), the coefficients of which are doubly- y Google 458 ANALYTICAL FORM OF INTEGRALS [146. periodic functions of the second kind, with and 0' for multipliers. In particular, the integrals have the form X, » = *-,(.) + /.*■,(«). x,W-*'.W + -r.-*'.W+W-f.« + K.*',(4 and so on. The functions F are doubly- periodic of the second kind with factors 8 and &; /, is a polynomial of the first degree in u (s) and v(z); /a and J^ are polynomials of the second degree in the same quantities, having I^ as the aggregate of their terms of the second degree ; /j is a polynomial of the third degree in the same quantities, having I^ as the aggregate of its terms of the third degree; and so on. To prove this, we note in the first place that </>i {z} is a doubiy- periodic function of the second kind with the multipliers 6 and ff. As for Xi (■2)1 ■'^e have ?6(g + <^) _ Xi(£) , ^ If therefore we take the function p,, (z), where ?.,(-). J. W + j1«W, we have *(» + . iy-M'*" W and therefore ^ »(» + «') -ft.(2> -P21O is a doubly -periodic function of the first kind. Let -^2(2) denote the product of this function and 0i (z) ; then F^ (z) is doubly- periodic of the second kind with multipliers and 8', and we have y Google 146.] ASSOCIATED WITH MUtTIPLE EOOT 459 Similarly, if ...«^{i(ty--}.w.|i(|)"-^}«(4 we find to be a doubly- periodic function of the first kind. Let F^{z) denote the product of this function and 0i(3); then Fi(z) is doubly-periodic of the second kind with multipliers and &, and wc have X. (^) - ^. W + p. (') f. W + iiP=." (^) - r- (')! * W- Similarly, after reduction, X, (.) = -F. W + p„ » ^, («) + (ip.- W - ft, (2)} -f. W + (iP.'W-p..W!*.W where -Fj (z) is a doubly-periodic function of the second kind with multipliers 6 and $', pt, (s) is a polynomial in u and v of the first degree, and p^ (2) is a polynomial (not homogeneous) in u and v of the second degree. And so on, in general : the theorem is thus established. Construction of Integrals that are Uneeokm. 147. Further progress in the efifective determination of the analytical forms of the integrals on the basis of the foregoing properties is not possible in the general case. When particular classes of limitations are imposed upon the coefficients in the original differential equation, such progress might be possible : but it frequently happens that some more special method loads more directly to the solution. The simplest case is that in which the equation possesses a uniform integral, or in which the equation has several uniform integrals: but, of course, the preceding investigations in §§ 141 — 146 apply to all equations of the type considered, whether they have uniform integrals or not. When all the integrals are uniform (and this can be determined independently by consider- ing their forms in the vicinity of the singularities), then the y Google 460 EQUATIONS HAVING [147. doubly -periodic functions of the second kind arising in the pre- ceding investigation are uniform functions of z; and a general method of constructing such functions is known*. Instead, how- ever, of using the preceding results, it sometimes is more con- venient and more direct to infer the irreducible singularities of the integrals from the differential equation itself. These are used to construct an appropriate uniform doubly-periodic function of the second kind; the remaining quantities needed for the precise determination of the integral are then inferred by substituting the expression in the differential equation. Ea:. 1. Consider the equation t with the usual notation for the Weierstrassian elliptic functions; q and ^ are Constanta. The only irreducible singularity that an integral can have is 2=0. The iodicial equation for e=0 is m(»-I)(«-2)-6«=0, the roots of which are -1, 0, 4; and the expansions that respectively corre- spond to the roots are easily proved to be Wj-l-l- 1^(33^-1-^503!^ + ..., Thus no It^arithniB are involved ; every integral is a uniform function of z, being of the form. Aw^-\-Bw^-\-Ow^; and at least one integral of the equation is thus a uniform doubly*periodic function of the second kind. We proceed to its construction. This doubly-periodio function of the second kind cannot be devoid of poles, if it is to involve the first of the above integrals in its expreasion. (If it were devoid of poles, it would alsoj be devoid of zeros in the finite part of the plane : and then {I.e.') it could only be an exponential of the form e", which is manifestly not a solution of oiu' equation.) It has one irreducible pole ; it therefore has one irreducible zero in the finite part of the plane. Let the latter be denoted by - a, which at present is unknown. We now consider § the elementary function " (■' + «) M * T. J., g§ 137—139. + It ia a modified form of an eiiuation given by Picai'd, C'i'dle, t. kc, p. 290. t T. F., i 139. g T. F., U. y Google 147.] UNIFOKM PERIODIC INTEGRALS 461 which has £=0 for an irreducible simple pole, and -a for an iiTeducible aiinple Kero; its oxpansion begins with s~i, and the function muat therefore agree with the integral above obtained in the vicinity of 3 = 0. (The constants X and a determine, or are determined by, the multipliers of the periodic function ; but at present these are unknown, and so X and a must be detormined in another maimer.) To expand the above function in powers of 3, we have the Weieratrassian functions on the right-hand aide being functions of a. »(.)- -J+(>.+0+i'{(»+0'-B+i''»+0"-3(i+f)»'-rl +£»+o*-8(^+f)'f-"ip+nf -*■+•«)+•■■ This is to satisfy ttie dilferential equation, so tbat it must be of tlie form Clearly J = 1, iJ = X + f, for this purpose: the value of P would be needed for the complete expression, but we merely require X and « at present. Com- paring the coefficieutB, we thus have ^ = 1, a-x+C, ^— (x+O'-f, so that X and a are determined by the equations (X+O'-J'— 1 (x+o'-3(x+f)t>-p'.jfir -•V x+f. _ where a is determined bj' the relation Sjy + (3a>-j,)J> + J|3'-a'-J,.0. The function on the left-hand side is a doubly-periodic function of tho first kind: it has a single irreducible pole, which is at 3=0 and is of multiplicity three. Hence it has three irreducible aeros, say a^, a^, a^; and their sum is congruent to 0, bo that we may take a^ + a^ + a^^O. y Google 462 EXAMPLES [147. In general, u^, Oj, «j ure unequal, because a and /i are general constants; the discussion of the critical conditions, that lead to equalities between a,, a^, a^t and of the consequent modifications in the complete primitive, is left as an K- S W+ a„2g)(a,) ' ('-1.2,3), ^^^)^ is an integral of the equation for each of the three values of r. Tho primitive of the equation is whore A-^, A^, A^ are arbitrary constants. Jix. 2. Obtain the relations which express the integrals w^, w^, v>^ of the equation in the preceding example in terms of H";, W^, W.^; and determine the multipliers of the integrals. Ew. 3. Obtain the primitive of the equation in the form w=Ae<^'^ + Bf{z). Ex. i. Verify that the primitive of the equation da''^ dn. y=^cos(nam^)+£sin{»iam:^). Ex. 5. Prove that, if / be an odd function and J^ be an even function, both doubly -periodic in the same periods, the integrals of the equation rfe= ds oan bo expressed in tenns of JJ (z). Hence (or otherwise) integrate the equation cr ' r J^x. S. Determine tbe relations among tlie constants (if any) in tlie equation »'"+(a-SB»'+(y+(iJ)-|j)>.0, in order that every integral of tlie equation aliould te uniform ; atid assuming tile relations satisfied, shew that the equation has three integrals of the form '(■+") ^ y Google 147.] EXAMPLES 463 Ha. 7. Shew that the equation has an integral of the form ^""^ ®(^) ^"^ "' provided 3a + / = 3(l+i2). and that it then has three integrals of that form. Obtain these integrals. (Mitte^-LefBer.) Eu-. 8. Obtain the integral of the equation in the form ^- ■&(:.) where the constants X and « ai'o given bj the equations A-(14-(t=) + 3(A=-i^8na«) = 0, 2X=-6Xi"8n^B + 3X(H-F)-4i2sn«)cnQ.dnco-Ai=0. Verify that, in general, three distinct integrals are thus obtained. (Picard.) A'iB. 9. Prove that the equation has an integral of the form ^ &(ic) ' provided 2a + S = 8(l+^)i and that, if this relation he satisfied, it has fonr such int^rals. Obtain them. (Mittag-Loffler.) Em. 10. Verify that the equation (m-.-.,l'<.)g-2.n>:«oxdn^J + 2j|l-2(I+i').n-,. + 3f .■,•.).! hiia an integral of the form provided an^'ci-sn^B ' afc2sn*K-2(I+i=)sn«a + l ' and obtain the primitive. Hence integrate the equation where A is a constant y Google 464 lamp's [147. Ew. 11. Discuss the equation S + |-2J,(.))*„J + ,, + ,p(.,,..„, for those cases when every integral is a uniform function of z. Ea:. 12. Shew that there are three sets of values of the constants o, and 6, for which the equation 33-(Ar"(»)+ir (»)+"?(")+»)? admits aa an integral a uniform douhly -periodic function ; and obtain the integral (Math. Tripos, Part ii, 1997.) Ex. 13. Prove that the equation y'"-2»(K + l)y'p(2)-2w(ft + l)/^'(.) where a ia an arbitrary constant and n is a positive integer, has a uniform function of z for its complete primitive. (H^lphen.) Ex. 14. Construct the equation which has for its complete primitive and, for a properly determined value of /(a), is devoid of the term in -^ . Likewise construct the equation which has w={a,+agsn3 + 0!3cnj+aidns}/(j) for its complete primitive, with the corresponding determination ai f{z) to remove the term in -^ . In each case, the quantities a^, a^, %, a^ are to be regarded as arbitrary constants. (Halphen.) Ex. 15. Prove that the primitive of the equation is a uniform function of z, when m is an integer multiple of 3 ; and discuss the primitive, when the integer n is prime to 3. (Halphen.) Lame's Equation. 148. One of the most important instances, in which a dif- ferential equation with uniform douhly-periodic coefGcients has a uniform doubly -periodic function of the second kind for its integral, ia Lamp's equation or, rather, the more general form of y Google 148.] EQUATION 465 Lamp's equation as discussed in the investigations of Hermite, Halphen, and others. The form used by Hermite* is where n is a positive integer and .S is a general constant; the form used hy Halphenf is with the same significance for n and B. We shall use the latter form of equation ; it is selected for convenience and for its slightly greater generality owing to the functional independence of the The mode of discussion is the same for the two forms. As we are concerned with the application of the general theory!, rather than with the special properties of the functions defined by Lamp's equation, only an outline of the solution of the equation will be given here. The detailed developments, and references to further memoirs, will be found in the authorities just quoted. It may bo not without interest to indicate how this form of equation arises from the equation da? dy' d^ ' characteristic of the potential in free space. When orthogonal curvilinear coordinates a, /3, 7, as defined by three orthogonal are used, then the equation becomes do:[BC Sa>'^Sg\CAafil St\AB dy) • * " Siir que <i e apjl at on deE foBctions elliptiques, " a separate repiitit (1B85) Irom the C np He E s + Traite dts to ctio e Ipt q es t ch. xir. J That is the theory of the un f rm doubly -periodic functions of the second kind which are ntegials of tlie differ nt al equation. It has been proved (g 54) that, by an app 01 t anal in on the equation can be changed eo as to be of Fuehsian type I. IV. 30 y Google 466 SOURCE OF [148. where «-©■-©■-©■• Choose, as the orthogonal surfaces, the three quadrics which are coiifocal with a given ellipsoid ; and let \, fi, v be the roots of the equation ir} if ^° _ -I a cubic in 9. Then* we take ([2 + \=:^(a) - ^1, ¥ ■\-\^f{oL) - e.,, c^ + \ =g>(a) -- e^, «' + ^ = ^(^)-e„ 6^ + M = F(/3)~e.„ c-^ + ^ = g)(^)-e„ Now jd\\^ /dXV /3XV Kdcc) ^ \dyl ^ \dz} - ^'^: where \ _ g? y s^ jiT" ^ (a' + Xf + (6" + X)" "'" (C + X)' (X-y)( X -») ((•■ + X)(6" + X)(c" + X) !>■■(«) 1 „, 1 Similarly so that the equation for the potential becomes * Greenhill, Pi-oc, ioiid. .Viif/f, Soc, t. xviii (18B7|. p. 27S. y Google 148.] lamp's kquation 467 or, what ifi the same thing, IP(«-S>(7)lj„'^+l!'(7)-?WlU+[S>W-p(ffll|^ = 0. For the purposes contemplated in the transformation, the quantity V is the product of a function of a, a function of /3, and a function of 7, or is an aggregate of such products ; and it is a uniform function of its variables. Hence, writing F=/(»)!,(/3);.(7), where /, g, h denote uniform functions of their arguments, we have where w =f when z = a, w = g when z=^^, w = h when s — y, and A, B are constants independent of a, /3, 7 : they must be such as will, if possible, make v) a uniform function of its argument. The only possible singularities of w are ^ = and points congruent with z=0; hence, after the earlier investigations, we consider the irreducible point 2=0. The form of the equation shews that it will be an infinity of w; and thus it must be a pole, say of order n, where n, is, & positive integer. Thus we have, in the vicinity of the pole, «; = J + -,^^,+ ... =^-K(s), where R (2) and iJi {z) are regular functions of z, such that BA£)_ Hence, in the vicinity ni z — 0, we have - -j-^ =— -^ — ^ — ' (1 + powers 01 z), y Google 468 INTEGRATION OF [14S. and therefore ^-.(n + l). a limitation upon the form of the constant A. But there is no limitation upon B, necessary for the existence of integrals of the type indicated ; and therefore the differential equation may be taken in the form as stated. To obtain Hermite'a form, we write , , _ «! — 63 _ - ,^ J _e.^ — e^ f {s) - 03 - gnSy ' V- ^(^1- ^3>. ■ - g^ _ g^ • as usual, and then take y = cc + i'K' ; the equation becomes where B" is a constant. 14!). The method of solution of the equation is based upon the knowledge that there is at least one integral in the form of a doubly-periodic function of the second kind ; the limitations, that have been imposed upon the equation, secure that this function is uniform. Moreover, the integral has only one irreducible pole, viz. at s = 0, and the pole is of order n. There are two modes of using these results in order to construct the integral. By one of them, we use* the further property that a uniform doubly- periodic function of the second kind has as many irre- ducible zeros as it has irreducible poles, account being taken of the orders of the points in each category. Accordingly, in the present instance, the integral has n irreducible zeros : let them be — rii, — «a, .--, — <^ij- Consider the uniform function which is doubly- periodic of the second kind; its (single) irre- ducible pole is of order n and is at s = 0; and it possesses the * r. F., %% 139, 141. y Google 149.] lamp's equation 469 necessary « (unkiiown) irreducible zeros, so that it is of a suitable form. We have 1 __ w dz In order to simplify the right-hand side, it is conveoient to take and so Hence 1 d?w 1 idwy , , o / , and therefore 1— _x+i I y'w^ p'w li)(«,)-yW( .lii y'(n,)-p'W p'(a, ) -y'W Tlie first term on the right-hand side is equal to To modify the second term, where the summation is for pairs of unequal values of r and s, we have li>'('V)-y'W P' (»■)-»'(») (>("»)-pW ■ «>(«.)-«'(»> _ yW- g.pW-g.-p'(»r)y'(''.)-f'MI»'K) + t''(».)l after easy reductions, where i,. ?(«)-!>("-) PW-PK)) . P' (»>) + P' (a. ) . ' P(»r)-S)(».) ' y Google 470 lamp's equation [149. and thus the second turm in the expression for —^\-j-] becomes »{,.-i),,(.)+2(..-i)_^ls.(«.,)+i_|f'<|>f^'l;i,,. where the summation is for all values of 5 from 1 to w except only s = r. Then ^ | y(a..)-y-W {f,^ aJ, mtW-fM U-i ) Comparing this result with the differential equation under con- sideration, we naturally take i' L,, - 2\ for all values of r, that is, »'M + f'('h) P'W + f'l". ) ^ _ 2^^ j>((i,)-i)(ii,) J)(o,)-|>(o,) f'W + y'W , y'(».) + t^(».) , _ 2x fW-|>(«.) «>(".)-«><».) f'(°~) +»'(■■■) ^. g'W + y'W ^ _ 2^ «'(».)-8'(»i) «'("•)-(('(«!) and (2" - 1) ^ g* («.) + X^ = 5. Adding the n former equations together, we have = 2n\, so that \ vanishes. Hence if the n quantities di, a^ a„ o.re determined by the equations y'W + y'W y'W + »''('■.) ^ _ „ (f'W + f'W , y'(».) + y'(a.) ^ ^ D ?(".)-(»(»■) fW-pW {which are equivalent to only k — 1 independent equations, becaii^e the sum of the n [eft-hand sides is zero) and by y Google 119.] XNTEGKATED thm F{z). a{z + a^)„{z + a.i...<, »-w is an integral of Lamp's equation i^ = «(« + l)^(2) + B. w dz^ ' The equation remains unchanged when —2 is written for z; hence F{—z) is also an integral. Save in the case when the constants a are such that F{z') and F{-z) are effectively the same function, we have two independent integrals of the equa- tion, which therefore is completely solved. 150. Another method of arranging the necessary analysis is as follows. Consider the equation where F(z) is a doubly- periodic function; by Picard's theorem (§ 142), an integral (say Wi) is known to exist in the form of a doubly-periodic function of the second kind. If then we write _ 1 dwi the quantity w is a doubly-periodic function of the iirst kind ; and it satisfies the equation The irreducible poles of w,, in their proper order, are known from the singularities of the original equation ; let them be jt in number, account being taken of multiplicity. Then each of them is a pole of v, of the first order ; and the sum of their residues for vis —n. The number* of irreducible zeros of w, is also n, account being taken of multiplicity; each of them is a pole of v, of the first order, and the sum of their resid\ies for v is +n. We therefore construct a uniform doubly -periodic function of the first kind, having these poles, all simple, viz. the known poles arising through the singularities of F, and the unknown poles y Google 472 EQUATIONS HAVING [150 arising through the zeros of Wi, taking care to have —n and +n for the respective sums of the residues. The general expression for such a function is known*: when substituted as a trial function in the above equation, comparison of the results leads to a determination of the constants. As an illustration, consider the equation 1 iPw „ , , T. The irreducible pole of g) (e), viz., 2 = 0, is the only irreducible pole of Wi, and it is of the first degree. Accordingly, it is a simple pole of v, with a residue —1. Further, there is (by the preceding argument) only one other pole of ii : it is simple, and has a residue + 1, As w is a doubly- periodic function of the first kind, an appropriate expression is .i:(«-c)-fw + f(c) + i., say ; and b, c have to be determined by substituting in the equation * + .. = 2p(.) + a Now dv , , / i and by the addition- theorem for the f-function, we have '' + *j)W-j.(o)- Substituting, we have if W + f(c) + b- + b ^>-+ ^'l-) = 28, («) + B, which must be satisfied identically. Accordingly, 6 = 0, ^(c) = B; and thus, with a known value of c, ^ = j:(2-c)-r(5) + f(c), * T. P., S 138. y Google 150.] DOUBLY-PERIODIC COEFFICIENTS 473 SO that There are two values o£ c, equal and opposite : the construction of the primitive is immediate. Ex. 1. Shew that two independent integrals of tiie equation ill the case when 5 = ej, are given by |y(.)-,,(', (»>(.)-.,}»{(■(.+.)+.,.)! and obtain the integrals in the cases, when B^e^, and B = e^, respectively. K:'). 2. Obtain the primitive of the equation (where q i.s constant), in the form where DiacuaM the solution in the three particular cases <i = I+F, 1, lc\ (Ilermite, Ex, 3. Shew that satisfies the eqviation if ^ ("i)' P ("2)1 ^ ("3) "^^ *hs roots of the oubic equation and deduce the primitive. (Ilalphen. Ex. 4. Shew that the primitive of the equation can be expressed in finite form for appropriate values of the constant B it the following cases : — I. When *i is an even integer, =2»i, then either where e^^, e are any two of the throe constants e^, ^j, e^ : y Google 474 EXAMPLES [1 50. 11. when )t is an odd integer, =2(1? — 1, then either «'={PW-«;.}*{c™_,iy"'-'(^)+...+C„}, where e^ is any one of the three constants e„ e^, e^. Determine the number of sohitions of the specified kind in each of the cases indicated. (Crawford.) Ex. 5. Shew that an integral of the equation where h is a coJiatant and m, n are integers, can he expressed in the form _,»,(»-.,) a, (.-.,).. .8, fr-....) z zK) in the usual notation of the theta-ftinctions, ^,, z^, ..., ^m-m hcing appropriate constants. Obtain the primitive. (M. Elliott.) Bx. 6. Obtain the primitive of the equation »-4Ji%)<*«"<''-''»*W-« Ea:, 7. Shew that there are two values of ij, for which the equation i5-i.+i,J>»-2~r(') + t""S''». where m is a constant, possesses an integral of the form ^ ,-.-t.;(.|.(«-i'). ,(■) ■ and, for each such value, obtain the primitive. (Bonoit.) -fie. 8. Shew that there are m+l values of k^,, for which the equation where m is a constant and n a positive integer, possesses an integral of the Prove abo that, if the right-hand side of the differential equation be increased by * (s), where * is a doubly-periodic function of the first kind having all its poles simple, a corresponding theorem holds as regards the integral, if it,, be properly determined. (Benoit.) y Google 150.] ALTERNATIVE PROCESS 475 Es. 9. Integrate the eqiiation where n, n', n", n ate positive integers. (Darboui.) 151. The other mode of utilising the known properties of the integral, when it is a uniform doubly-periodic function of the second kind, is to obtain the actual expansion of the integral in the vicinity of its irreducible pole and thence to construct its functional expression in terms of the elementary function »(,) ■ where a and \ are initially unknown constants. Some indication of the process is given in Ex. 1, § 147; but a slightly different form will be adopted for the present purpose. We take the elementary function in the form '{' + where p and a are now to be regarded as the constants to be determined. The expansion of this function in the vicinity of its irreducible pole at s = is + sV {p' - 6/>=F («) - ^99' («) - W («) + *?=} ^ + ■ ■ ■ ■ If, in the same vicinity, an integral of the diiferential equation exists in the form . . . + ' + (to + positive powers, then we rnay take where a comparison of expansions serves to determine the con- stants a and p. The integral thus is known. y Google 476 EXAMPLE [151, An illustration will render the details clearer. In the case when 91=2, the equation is 1 d'^J) „ , , n Let 1 a, w = -^ + - + (ti + ctjs + a,s^ + . .. be substituted in the equation ; we find a,= Q, ((,, = 0, a,-0, ... a,- -IB. a, = i-,]P-i^g„... so that Manifestly, the form to take is __dG_ dz ' and then eompai-ing the two expansions, we have -ilp'-p(«)l--i-B. /-3(jy(»)-p'(») = 0. These equations give BM--27y. 3y'(a) The former in general leads to two irreducible values of a ; the latter uniquely determines p for each of these values of a. Denoting the two values of a by a and — a, and writing <r(«)<r(o) the primitive of the differential equation is ... r<ie. . ..dc. y Google 151.] EXAMPLES 477 Bx, 1. Discuss tho integral of the equat.ioii when a, as obtained in the preceding solution, has the values 0, <u, m', lu", respectively. Ex,. 2. Prove that an integral of the equation in given b, the constants p and a being given by the equations Deduce the primitiv' y Google CHAPTER X. Equations having Algebraic Coefficients. 152. The differential equations, considered in the preceding chapters, have had uniform functions of the independent variable for their coefficients. We now proceed to consider (but only briefly) some equations without this limitation : one of the most important classes is constituted by those which have algebraic functions of the variable as their coefficients. For this purpose, let y denote an algebraic function of the independent variable X, defined by the equation where 1^ is a polynomial in x and y, and the equation is of genus p. With this algebraic equation we associate the proper Eiemann surfiwe of connectivity 2j) + 1. We assume that the linear differential equation has uniform functions of x and y for its coefficients, so that each of these is a uniform function of position on the surface : and we write the equation in the form '""" + a, (a^, y) ;^ + a, (^, !/) ^:_~ + . . . + a,„ («^, ;/) ^^ = 0. rf^™-^"=^-^-^^rf^' „=Jf :, y) dx the exponential in the factor of u on the right-hand side being an Abelian integral ; then the equation for w is y Google 152.] ALGEBRAIC COEFFICIENTS 479 devoid of the derivative of order m — l; all the coefficienta Pa, ■--. Pm ^>^^ algebraic functions of x, and are uniform functions of X and y. This is the form of equation which will be discussed. Let (iCj, jio) denote any position on the surface, which is not a singular point on the surface and in the vicinity of which each of the coefficients P is regular. Then an integral exists, which is regular everywhere over a domain in the surface, and is uniquely determined by the assignment of arbitrary values to iv and to its first m — 1 derivatives at (ic,,. po)- I" f*ct, all the results relating to the synectic integrals of an equation with uniform coefficients hold for the present equation in the domain of (x^, y^. Next, let account be taken of the singularities of the equation ■>|r = and of the associated surface. As these affect all the coefficients of all differential equations of the class considered, and thus afford no relative discrimination among the functions defined by those equations, we shall assume them simplified as much as possible before proceeding to consider the properties of the functions. Accordingly, we shall suppose that, if the equa- tion -"It = {or the Riemann surface associated with it) possesses a complicated singularity, it is resolved* into its simplest form by means of birational transformations, so that we may write where ff and h are uniform functions which, in connection with ^ = 0, admit of uniform expressions for f and 17 in terms of «— e and y—f, and are such that ^=0, »j = is an ordinary position on the transformed Riemann surface. The positions on the surface, that have to be considered in connection with the differential equation, are now ordinary positions : and therefore, in dealing with the theory of the equation, no generality is lost if we assume that the singularities of the equation 1^=0 and of the Riemann surface are ordinary positions for the integrals. (Of course, in any particular example, it may happen that a multiple point on the curve ^ = 0, or a branch-point of the associated surface, is definitely a singularity of the equation. In order to discuss the nature of the integrals in the vicinity of such a point, we takef yGoosle 480 FUKDAMENTAI, SYSTEM [1,52. where ]) and q are integers, and S is a holomorphic function of its argument that does not vanish when f=0; and then we investigate the character of the integrals in the vicinity of f-O.) Lastly, let (a, b) denote a position on the Riemann surface (being a pair of values given by the differential equation) such that the coefficients of the equation are not regular in the immediate vicinity of (a, b) ; after the preceding explanations, we may assume that ?/ — 6 is a holomorphic function of x — a in the immediate vicinity of the position. The character of the integrals in that region is determined, after substitution oi y — h in terms of x — a, in association with an indicial equation ; and the general processes of the theory, in the case of differential equations with uniform coefficients, are applicable to the integrals in the vicinity of {a, b). As in that earlier theory, we have a fundamental system of integrals existing at any ordinary position on the surface, the system being composed of m linearly independent members. Continuation of these integrals is possible : and by taking all admissible paths iirom one ordinary position to any other ordinary position (care being taken to avoid the actual singularities), and assuming an arbitrary set of initial values at the first point, we shall obtain all possible integrals at the second point. Similarly, by taking all possible closed paths on the Eiemann surface, which begin at an ordinary point (.x^, ya) and return to it, we obtain new integrals at the end of the path ; and each of these integrals is linearly expressible in terms of the members of the initial fundamental system. A Fundamental System op Integrals, and the Fundamental Equation. 153. Let Wi, Wi, ,.., Wm denote a fundamental system at an ordinary position («o, y„); and let the variable of position describe a closed path on the surface returning to (a:„, 7/0), this closed path being chosen so as to include the singularity (a, h) but no other singularity of the differential equation. Suppose that the effect upon the fundamental system, caused by this variation of the variable of position, is to change it into y Google 153.] OF ISTEGRALS 481 w^, w^, ..., Wm : then, as in the case of uniform coefficients, the latter set also constitute a fundamental system, and the two systems are related by the equations Mj/= 2 a^^w^, (m = i, ni\ where the determinant of the coefficients a is different from zero. This determinant is (as in § 14) equal to unity. For let A denote the determinant of the fundamental system and let A' denote the same determinant in relation to the fundamental system w' ; then, if A denote the determinant of the coefficients a^^, we have A' = ^A. Now, because the term involving the {m is absent from the differential equation, wi — l)th derivative of w have, as in § 14, where C is a constant. Let the function A, which is equal to C in the vicinity of (a:^, y^, be traced along the closed path which the variable of position describes on its return to {xa, y^\ it is steadily constant, and its final value is A', so that ^ = 1. Further, as in the cases when the coefficients are uniform functions of the independent variable, it is possible to choose a linear combination v of the members of the fundamental system such that, if if denote the value of v obtained by making the variable of position describe the afoi'esaid closed path, we have 31 yGoosle 482 FUKDAMENTAI. KQUATION [153. The multiplier ^ is a root of the equation «2i , Hsa — ^ flam = 1 + I,B + 1,0' + ... + i^-,^-' + (- l)-"^"" = 0. This equation is independent of the choice of the fundamental system, so that its coefficients may be regarded as invariants of the linear substitution, which the fundamental system undergoes in the description of the closed path round («, b). 154. If some, or if all, of the integrals in the vicinity of (a, b) are regular in the sense of § 29, then an indicial equation for the singularity exists; and if p be a root of this equation for an integral with a multiplier 8, then If no one of the integrals is regular, there is no valid indicial equation. In the first case, the general character of an integral is determined by the value of p : and the explicit form is obtained by substituting an expression of the appropriate chai'acter so as to determine the coefficients. In the second case, various methods* for obtaining the value of S have been suggested, by Fuchsf, Hamburgerj, and Poincare§; the most general is the method of infinite determinants, due to Hill and von Koch, and expounded in Chapter Vili. Without entering upon details, it may be said briefly that many of the properties of linear differential equations having algebraic coefficients can be treated by processes that, except as to greater complexity in the mere analysis, are the same as for equations with uniform coefficients. It therefore seems un- necessary to discuss them at any length, as they would lead to what is substantially a repetition of a discussion already effected for less complicated equations. ■ Seo g 127. t Creiie, t. lxsy (1873), pp. 177—223. J CrelU, t. LKXKiii (1877), pp. 185-310. § Acta Math., t. iv (1881), pp. 208 et aeq. y Google 154.] EXAMPLE 483 A systematic discussinn of equations having algebraic coeffi- cients and development of many of their chaiacteristie properties will be found in a series of memoirs by Thom6*. Ex. 1. Consider the equation where the variable y is dciined by the relation ' +y=i and a, a, b, c are Luiistant^ The position, at infiiiitj i& a singuliiitj of the difleiential equation in ©aoh of the two sheets ot tlie Eiemann suifice The integrals aie regular in that vicinity in one oheet ^nd the exponents to which thej belong ire the roots of provided o + 6i is not aero ; but, if a + bi=0, the integrals are irregular at infinity in that sheet. Similarly, they are r^ular in the vioinity of infinity in the other sheet, and the exponents to which they belong are the roots of '<'+'>+5^-K).-''' provided a — bi is not zero; but, if a — iii = 0, the integrals are irregular at infinity in that sheet. The other singularities of the equation are given by ax+by-VC=0\ When these are distinct hma one another, let them be denoted by ; =cos fl, y = sm6; x=aoa(j>, y=3mi^ The integrals are regular in the viuimty of each position; and the leajectue mdiciil equations are ''"■^" + („ S°c.t«/ -° '''-"■^ (»-i°c.« <-°' When the two singularities coincide, let the common position be denoted by a:=c<m\lf, y — sirt'jr ; and then In the vicinity, we have ar=cos-J,+a , = .int-|cot^-i_^!^ + jC^-..., * Crelle, I. cxv (1895). pp, 33—33, 119—149 ; ib., t. csix (1898), pp. 131—147; ib., t. oixr (1900), pp. 1—39; ib., t. cslsii (1900), pp. 1—39. y Google 484 EXAMPLES OF EQUATIONS HAVING [154. so that the equation is The integrals are not r^ular ; but the equation may have one normal integral, and can even have two normal integrals, of the type aUin V ! ■where / is ft polynomial in ^. The forms, and the conditions necessary to significance, can be obtained as in §§ 86 — 87. Ex. 3. Discuss in the same way the singularities of the same differential equation, when the irrational quantity y is given by the respective relations (i) ^+f = \, (ii) /-4i^-yja^-^3. Ex. 3. Let w, and a, denote a fundamental system, of the equation in Es. 1, for 3' = (1— a^)^ ; and let «, and r^ denote a fundamental system of the same equation for y—-(l-a^)*. Shew tliat the linear equation of the fourth order, which has Mj, it^, Vj, v^ as its integrals, has rational functiona of x for its coeffioieiits ; and obtain them. Ex. 4. The equation has its primitive in tlie form It is natural to inquire whether an equation £^-^,. can have an integral of the type where nr (a:, y) is a rational function of a: and y. A general method for such aa inquiry has been given by Appell*, thoi^h it is not can'ied to a complete issue as regards detail ; it will be sufficiently illustrated by means of the equation iP"W_ a x-^^y _ where a?-^y^ = \, it being required to find under what conditions, if any, the equation can have an integral of the form y Google 154.] ALGEBRAIC COEFFICIENTS where nr (,«, y) m s, rational function of x and y. Since We assume that each of the quantities a+flt, a+hi, is different from nero. By adopting the method in the precediug Ex. 1, the integrals of the equation in w are easily aeen to b« regular in the vioinity of a; = co , so that they have the form where ni-] is a holomorphic function for large values of x, not vanishing when :e=co ; and thus Substituting in the equation for or, we have Now the infinities of w are included among the points (i) :e= tc , which has juafc been considered ; there are two jwsaible values of X in each sheet : (ii) y = 0, with .r = l, ^=-1, which are the branch-points of the surface : (iii) ax-\-hy = <i, in each sheet. Moreover, the zeros of w are uuknowti from the differential equation : but they must bo considered, becaiise each of thetu gives a pole of la. Let such the number of suoh points being unknowiL All these points, whether infinities of w or zeros of w, can be singularities of -m. As regards the branch-points (ii), we may take y = 'i, ^=I-iiH..., in the vicinity of 1, 0, where jj is small ; and then so far as the governing term in ct is concerned. If tiiis be y Google 486 EQUATION OF SECOND ORDEIl [154. where ft>0, then n+2=2n, A'^+nA=0. Thus n=2; and we can have ^= — 2, or ^ = 0, as possible values, Similarij for the vicinity of — 1, 0. Next, at the two points (iii), where as+iri/=0, we have ^^^sinvf', y = coii\jr-, a tan 'jr = -- b. Then, in the vicinity, wo take ,K=^ain.;^ + |, y = oosi|.-^Uni^+.,., Thus the equation is d^ (.1.-8..) »1 de*' («■+»•)' e' su far as the gaverning term in cf is concerned. If this governing term be T .(»■+» ) 1 ' '- (»■+!.•)■ ■ Thus there are two possible values of rr at each of the two points. Lastly, as regards a point such as ,i;=/in the set (iv), it is easy to see that, if the governing term in ot be B (^ -/')•" 2n = n + ^, B'i = nB ; that 18, 11=1, and either B = \, 5=0, are possible values. This holds for every such point a:=f and in each sheet. Our required function luix, y), if it exists, is to be a rational function of X and j/t and we have obtained all the singularities that, in any circumstances, it might possess. We accordingly must take some combination of the possible iiifinitiea, which are j; = oo , with any of the values of X, ^-±l,y=0, with either A = -2,oi A=0, a3: + by = Qi, with any of the values of a, ^=/, with 5=1, or 5-0. y Google 154.] HAVING ALGEBRAIC COEFFICIENTS 487 A possible form is clearly a where (7 is a constant. We have (if this be admissible) a + bi ' from the first of the possible infinities : we take A = from the second ; then <-'='°"?' from the third ; and we take 3=0 from t!ie fourth. Hence we muat have a-bi for some possible values of X and of a- : that is, -{-gyp)'=^T-{'-<.->s>?'--«}'} the signs being at oiir disposal. Thi.s leads to a single value of ft via. and the condition ia satisfied by taking the negative sign on both siiies. We then have so that, with the above value of |3, an integral of the equation d^^ 2/iax-i-byy is given by Actual evaluation of the integral in the exponential can easily be effected. se, it would have been possible to discviss the particular equation by taking =l+(i' ^~i+(a' with ( as the new independent variable ; for the algebraic relation is of genua zero, and therefore* the variables can be expressed as rational functions of a new parameter. The new form of equation would then have uniform coeffi- cients. But the foregoing method, that has been adopted, ia possible for an equation ijr {x, y) = of any genua. * T. F., g 247. y Google INTRODUCTION OF [155, Association with Auxomokphic Functions. 165. It is manifest that some of the complexity in the analysis associated with the construction of integrals, either in general or in the vicinity of particular points, would be removed, if the equation could be changed so that, in its new form, its coefficients are uniform functions of the independent variable. This change would be secured, if both the variables x and y in the relation were expressed as uniform functions of a new variable z. Now it is known* that, when the genus of this relation is zero, both X and y can be expressed aa rational functions of a new variable z, which itself is a rational function of x and y. moreover, the expressions contain (explicitly or inaplicitly) three arbitrary parameters, which may be used to simplify the form of the resulting equation. Againf, when the genus of the relation is unity, both a: and y can be expressed as uniform doubly -periodic functions of a new variable z, while tg(z) and ^ {z) are rational functions of x and y ; moreover, the expressions contain (explicitly or implicitly) one arbitrary parameter, which again may be used to simplify the form of the resulting equation. And, in each case, definite processes are known by which the formal expressions of x and y, in terms of the new variable, can actually be obtained. When the genus of the algebraical relation ^{!e,y) = ^ is greater than unity, a corresponding transformation is possible by means of automorphic functions : not merely so, but such a transformation can be efl'ected in an unlimited number of ways. Further, it is possible to choose transformations that simplify the properties of the integrals of the diflerential equations to which they are applied. But, down to the present time, the instances in which the complete formal expressions of x and y have been obtained, and the application to the differential equations has been made, are comparatively rare. The results that have been * T. F., % 247. y Google 155.] AUTOMORPHIC FUNCTIONS 489 established are of the nature of existence- theorems. It is true that indications for the construction of formal expressions are given ; but the detailed analysis required to carry out the indica- tions is of so elaborate a charactet that it may fairly be said to be incomplete. The subject presents great, if difficult, opportunities for research in its present stage. A brief account, based mainly on the work* of Poincar^, is ail that will be given here. References to the investigations of Klein and others in the region of automorphic functions will be found elsewheref. The main properties of infinite discontinuous groups and of functions, which are automorphic for the substitutions of the groups, will be regarded as known. It is convenient to associate with any group a region of variation of the variable which is a fundamental region ; and for the sake of simplicity in the following explanations, it will be assumed that this region is such that, when the substitutions are applied to it in turn, the whole plane is covered once, and once only. Further, also for the sake of simplicity, it will be assumed that the axis of real quantities in the plane is conserved by the sufetitutions of the group. There are corresponding investigations, which establish the results when these assumptions are not made; but, as already indicated, the results are mainly of the nature of existence-theorems and cannot be regarded as possessing any final form, so that the kind of con- sideration adduced will be sufficiently illustrated by dealing with the simplest cases. In order to deal with the most general cases, it is nece^ary to utilise the theory of automorphic functions in all its generality ; yet the subject still is merely in a stage of growth, being far from its complete development^, 156. It is known § that, if x and y be two uniform functions of a variable z, which are automorphic for an infinite * This work ia beet esponnded in his five valuable memoirs in Acta Matheviatica, t. 1 (1882), pp. 1—63, 193—294. ib., t. in (1883), pp. 49—99, ib.. t. iv (1884), pp. 201—312, ib., t. V (1884), pp. 209—278. + T. F., chapters isi, ixii. t The most oonseeotive account of the subject is to be found in Frieke und Klein's VorUmngen ii. d. TkeorU d, mttmiiorphen Functionen (Leipeig, Teahner; vol, I, 1897; vol. 11, part i, 1901). § T. F., § S09. y Google '490 AUTOMORPHIC [156. discontinuous group of substitutions effected on z, then some algebraic relation +(«,j)-o subsists between them. Conversely, if this algebraic equation be given, it is desirable to express the variables x and y as uniform automorphic functions of a new variable z. For this purpose, we note that for general values of x, the variable i/ is a uniform analytic function* of m ; but there are special values of x, being the branch -point 8, at and near which y ceases to be uniform. Now suppose that x can be expressed as a uniform automorphic function of s, say the fundamental polygon being such that the branch-point values of X cori'espond to its comers (or to some of them), which include all the essential singularities of the uniform function /(s). Then, when substitution is made in the above relation, it becomes an equation defining ?/ as a function of ^ ; so long as z varies within the poiygona! region, y does not approach those values where it ceases to be uniform, for they are given only by the corners of the polygon. Hence y becomes a uniform function-t* of s ; and as a; is automorphic for the group of the polygon, it is at once seen that y also is automorphic for that group. Further, suppose that at the same time there is given a linear differential equation of any order, in which the coefficients are rational functions of sc and y. In addition to the branch-points which may be singularities of the equation, it naay have a limited number of other singularities. Let such a singularity be x = a, y — h, where of course 'if {a, b) = 0: for the moment, the question of the regularity of the integrals in the vicinity is not raised. If the polygon is constructed, so that x = a corresponds to one of its corners which is an essential singularity of the group, then that corner is an essential singularity of /(a). Hence, when the differential equation is transformed so that z becomes the in- dependent variable, the original singularities no longer occur so long as 3 is restricted to variation within the fundamental polygon : they can occur only for the special values at the corresponding If, further, the function f(s) is such that no special y Google 156.] FUNCTIONS 491 singularities for values of s arc introduced for values of w that are ordinary points of the equation, which will be the case ii f'{z) does not vanish within the polygon, then all the values of z within the polygon are ordinary poiuts of the equation, and all the integrals are synectic everywhere within the polygon. The singularities have been transferred to the boundary of the s-region; and thus the variables x and y, as well as all the integrals of the given linear differential equation which has rational functions of x and y for its coefficients, can be expressed as uniform functions of z within the region of its variation. AuTOMouPHic Functions and Conformal Bepbesenta.tion. 157. The relation between the variable s and the function x=f{s) can be considered in two different ways, the analytical expression of the significance being the same for the two ways. In the first place, the relation can be regarded as one of conformal representation. Assuming for the sake of simplicity that all the singular values of x are real, consider the problem* of representing the upper half of the 3;-plane bounded by the axis of real quantities conformally upon a polygon in the s-plane, bounded by circular arcs and having m sides : this conformal representation is known to he possible. If its expression be then f'{z) must not become zero or infinite anywhere within the polygon, that is, for any finite values of x; for otherwise, the magnification would be zero or infinite there, a result that is excluded save at possible singularities on the boundary. It is manifest that the representation remains substantially the same, if the s-plane be subjected to any homograpiiic trans- formation where a'd' — h'd — \\ for this will merely change the polygon bounded by circular arcs into another polygon similarly bounded. * T. F., % 271. y Google 492 CONFORMAL REPRESENTATION AND [157. Hence, in constructing the function for the conformal representa- tion, account nmst be taken of this possibility; and therefore, as {'. 4 = B,«'l, where {'.'] is the Schwar ziai 1 derivative, we construct thi tion {z, 4 We hare' k«i=i5; 1- ^ + s^ o° . il(x), (S- -0)- + *:.- say, where the summation on the right-hand side extends over all the singular values a of a;; the interna! angle of the s-polygon at the corner homologous with a is a-rr, and the coefficients A^ are real quantities. If oo is an ordinary value of x, so that no angular point of the polygon is its homologuo, then = tA„ = XaA + lS(l-aa = 2aU»+Sa(l~a^). li' oD is a singular value of x, whicii has an angular point of the polygon as its homologue, with the internal angle equal to ictt, then the summations being over all the finite singular values of x. The number of constants is sufficient for the representation. In the case when oo is not the homologue of an angular point of the polygon, we have m constants a,, m constants a, and m con- stants A„, subjected to three relations as above; as all these constants are real, there are 3m — 3 independent constants. But, if _ a'-X + b" ^ c"'X + d" • where a"d" ~ ^"c" = 1 and the constants a", b", c", d" are real, then the upper half of the ic-plane is transformed into itself; hence the m constants a are effectively only m — 3 in number, and thus the constants in / («) are equivalent to 3m — 6 inde- pendent constants, which can be used to make a solution determ- ■ The whole iovestigation is due to Schwarn; see T. F., g 371. y Google 157.] AUTOMORPHIC FUNCTIONS 493 inate. On the other hand, to determine the polygon, 3m constants are needed, viz. two coordinates for each of the m corners and a radius for each arc : but these are subject to a reduction by 6, for the representation is determinate subject to a transformation c'^+d" where a'd' — b'c' = 1, and the constants a', b', c', d' are complex, so that there are six real parameters undetermined. The number of available constants is therefore sufficient for the number of condi- tions that must be s In the case when =0 is the homologue of an angular point, we have ni — 1 constants a, m constants a, and m constants A„, sub- jected to two relations as above; as ail these constants are real, they are equivalent to 3m — 3 independent constants. The re- mainder of the argument is the same as before ; and we infer that the number of constants is sufficient to satisfy the number of conditions for the conform al representation. It need hardly be pointed out that, thus far, the polygon bounded by circular arcs is any polygon whatever; it has been taken arbitrarily, and it does not necessarily satisfy the conditions of being a fundamental region suited for the construction of auto- morphic functions. 158. That polygons can be drawn in the a-plane, suited to the construction of autoraorphic functions in connection with a given algebraic relation i/r (x, y) = 0, may be seen as follows. For simplicity, let the polygon be of the first family*, and let it have 2n. edges arranged in n conjugate pairs ; and suppose that q is the number of cycles of its corners, each cycle being closed. The genus p of the group is given by 2p = m -i- 1 - g. When the surface included by the polygon is deformed and stretchetl in such a manner that conjugate edges are made to coincide by the coincidence of homologous points, then for each cycle in the polygon there is a single position on the closed " T. F., %% 203, 293. y Google 494 FUNDAMENTAL REGION [158. surface obtained by the deformation. This closed surface corre- sponds* to the Riemann surface for the equation ^}r {x, y) = 0, which also is of genus p ; and thus there are q positions on the surface. ea<;h associated with one of the q cycles. Each such position requires a couple of real parameters to define it; and thus we have 2§ real parameters. Equations, which are biration- ally transformable into one another, are not regarded as inde- pendent r and therefore the effective number of constants in ^ {^' y) = to be taken into account is 'ip — 3, being the number"!" of class-moduli which are invariantive under birational transform- ation. Each of these is complex, so that the number of real parameters thus arising is 6p — f). We therefore have to provide for 6p — 6 + 25 real parameters, by means of the polygon. In order that the polygon may be properly associated with a Fuchaian group, it must satisfy certain conditions. Its sides must be arcs of circles, the centres of which lie in the axis of real quantities. As it has 2n sides, we therefore require 2ji centres on that axis and tn radii, making 4ft real quantities in all ; but three of the centres may be taken arbitrarily, for the polygon now under consideration is substantially unaffected by a transforma- tion V ' cz + dl ' where a, b, c, d are real ; so that the total number of real quanti- ties necessary is effectively 4« — 3. They are, however, not suffi- cient of themselves to specify an appropriate polygon 1 for conjugate sides must be congruent, a property that imposes one condition for each pair of edges, and therefore n conditions in all : and the sum of the angles in a cycle must be a submultiple of 2-n; so that q conditions in all are thus imposed. Hence the total number of real quantities necessary is = 4m — S — n — q = Sn.- ^~q = 6p~6 + 2q, in effect, the same as the number of real parameters given. y Google 159.] FUCHSIAN EQUATIONS 495 AUTOMORPHic Functions and Linear Equations oB' the Second Order : Fuchsian Equations. 159. In the second place, the variable s, and the automorphic functions x and y, can be associated with a Unear differential equation of the second order. Let then it is easy to verify that v^ da? «s &3? IV ^ ' where (*', z\ is the Schwarzian derivative of as with regard to z, and 1^ = dxjd,z. It is a known property* that, if *■ is an auto- morphic function of z, tben the function is automorphic for the same group; hence it can ] rationally in terms of a; and y, where Denotiog its value by — 21, where / is a rational function of X and y, which may be a rational function of x alone, we have V] and Vj as linearly independent integrals of the equation ,-„ + /jf = ; the quantity z is the quotient of the two integrals. The analytical relation is effectively the same as before ; for if {z,x] = 2I, we knowf that z is the quotient of two integrals of n Difereiitial Kquatioi y Google 496 AUTOMOKPHIC FUNCTIONS AND [159. Moreover, 90 that the results agree in form. The difference is that, regard- ing the relation as a problem of conformal representation, we have been able to calculate the value of / in gi'eater detail than in the alternative mode of regarding the relation: but the con- siderations adduced in connection with the differential equation have been of only the most general character, and have not permitted any discussion of the form of /. When an equation of the form da? is given, where / is a rational function of x, or a rational function of two variables x and y, connected by an algebraic equation t(!l!.j)-0, it may happen that x and y are uniform fuQctions of z, the quotient of two integrals of the differentia! equation. But these uniform functions are not necessaiily, nor even generally, auto- morphic for a group of substitutions of s. Judging from the result of the consideration of the question as a problem of conformal representation, we should be led to expect that the constants, which survive in / after the conditions for uniformity are satisfied, might be determinable so that the uniform functions of z are automorphic. When this determination is effected, the equation is called* Fuchsian by Poincar^j if the group be Fuchsian. 160. We proceed to consider more particularly the properties of the equation ,-„-|-/)i = fl, in relation to the qiiotient of its integrals. Let jc = a, y = h be a singularity of the equation, where ■^{a,b) = 0; and let Limit [{x — ayi]it=a = p, so that the indicial equation for a is n(n-l) + p = 0. ' Acta Math., t. IV, p. 323. y Google 160.] FUCHSIAN EQUATIONS 497 Let n, and n^ be its roots, when they are unequal ; then two integrals of the equation are of the form and BO If atr be the internal angle of the jr-polygon at the angular point homologous with a, we must have and therefore that is, so that — 4p = 1-a the remaining terms being of index higher than — 2. This is valid, if a is not zero. When a is zero, ao that «i = % and therefore p — i, the integrals of the equation are jjj = (3;-([)«i[[l + ...} log (fl! -«) + powers oix — a], and so, in the immediate vicinity of a, we have 2 = — = log («! — »)+ powers ; and then the remaining terms again being of index higher than — 2. The quantity a, in terms of which the leading fraction in Z is expressed, depends upon the character of the singularity at (a, b). If the latter denote a singular combination of values for the equation then it is known* that the variables x and y can be expressed in the form ' T. F., § 97. T. iv. 32 yGoosle 498 FUCHSIAN [160. where S(^) is a regular function of f. which does not vanish when f = 0, and the expressions are valid in the immediate vicinity of the position. Let r be the least common multiple of p and q, and write then in that vicinity, we have {x-ar = z+..., SO that both x and y are uniform functions of s in the vicinity. The commonest instance occurs, when {a, b) is a simple branch- point ; we then have so that a = 4, If (a, b) he a singularity of some given differential equation of any order, say where i/r (x, y) = 0, three cases arise. Firstly, let all the integrals be free from logarithms, and let all the exponents to which the members of a fundamental system of integi'als (supposed regular) belong be commensurable ; then they are integer multiples of a quantity k~^, and we take ..l. (.-„, = ...... In that case, any integral is of the form = (a; — o)* R(a; — a) -''RC), SO that the integrals of the equation, as wel! as the variables w and y, become uniform functions of z in the vicinity of 2 = 0. Secondly, let the integrals (still supposed regular) of the fundamental system belong to exponents some of which at least are not commensurable quantities. We take ir = log(a!-a)-|-powers; yGoosle becomes integral of the form ix~arR(a:-a), i.e., a uniform function of s, valid for large values of \z\ : and this uniformity is maintained whether /j, is commensurable or not. Thirdly, let x = a be an essential singularity of one or more of the integrals, supposed irregular there. As in the laat case, we take 2 = log (ai ~a) + powers ; the integral may or may not become uniform for large values of U|. In the last two cases, if the expression for a: in terms of s, say be automorphic, then 2 = os is an essential singularity of the function f{s) ; and then, when z varies within the polygonal region, w does not approach the value a for which the integrals of the equation cease to be regular. Within the region, the integrals are unifoi-m. It is to be noted that the relation, adopted in the second case and the third case, woufd be effective in the first case also, so far as securing uniformity ; but the converse does not hold. The relation which, as seen above, corresponds to the vicinity of an angular point of the polygon where the sides touch, is the most generally applicable of all : the form of relation, corre- sponding to the first case, is applicable only under the somewhat restricted conditions of that case. 161. These conditions and limitations affect the quantity / in the equation for they determine the leading coefficient in its expansion near any of its poles ; but, in general, they do not determine / com- pletely. On the other hand, we so far have only secured the uniformity in character of the functional expression of x in terms of z: the automorphic property of the functional expression has not been secured. The latter is effected by the proper assign- ment of the remaining parameters in /. y Google 500 CONSTRUCTION OF [161, Aa a special instance, take the case in which the genus of the group and of the permanent equatii>n is zero ; so that, if the polygon has 2m edges, the number of cycles q is given by , = » + !. Taking the angulai' points in order as A„ A^, ...,A^a, and making the sides A,A, I A,A, ] ] A,^,A^ ] An ^„+,) A,aJ' A,„A^J---y A„+,A„J' A„+,A^J- to be conjugate pairs, the necessary n+1 cycloa are A,] A„A,^; A,-4™-i; ■-; A,„A„+,; A„+„ To define the polygon of 2n circular arcs, which have their centres on the axis of real quantities, we require the 4in coordi- nates of the angular points ; but these effectively are only 4k — 3 quantities, because the ^-plane is determinate, subject only to a transformation / as + b\ V ' cz + d)' where a, b, c, d are real. In each cycle, the sum of the angles is a submultiple of Stt : so that n + 1 conditions are thus imposed. Again, the edges in a conjugate pair must be congruent; so that n furthei' conditions are thus imposed. Accordingly, there remain 2ji — 4 real independent constants to determine the polygon. The polygon thus dofcermined defines a Fuchsian function; as the genus is zero, every function can be expressed rationally in terms of x, so that the equation for v (leading to s, as the quotient of two integrals) is -3- +lv=0, ax' where / is a rational function of x. Corresponding to the n + 1 cycles, there are w + 1 values of x ; let these be Let OtTt be the sum of the internal angles of the z-polygon corre- sponding to (X,, so that Uj. is the reciprocal of an integer; and take 0,1.^1 to be the quantity a for co . Then in the vicinity of a^, we have y Google 161.] FUCHSIAN EQUATIONS for each of the values of r. Thus, if we wribe and remember that I is a, rational function of x, we have «'«''=["]:.„■ for r=l, ...,n. In order to satisfy the condition for « = !», 6(x) must be of order 2« — 2, and aw- 1(1 -<■'«)«■"-+•... The number of coefficients in G (x) is 2m — 1 ; but the coefficient of the highest power is known, and there are n relations among the rest, owing to the conditions at CTi, ..., a„; lience there remain w — 2 coefficients independent of one another. Each of these is complex in general, so that they are effectively equivalent to 2w — 4 real constants. Assuming that the quantities Oj, .,., «« are known, it is to be expected that the 2« — 4 conditions for the polygon determine these 2n — 4 real constants. ; and we may take Os — O, In the simplest casc, \ ve have n ,= <h-- = 1, so that / = i a? ■-i(^ I)' The conditions for . X = 'K, give f + Hi- ".=) + J(l-< ) + = i (!-«.■). where a,, Kj, eta are the reciprocals of integers; the quantity / then is the invariant of the hypergeometric series. 162. As another illustration, which may be treated somewhat differently, consider the equation 1/' = a; (1 - ic) (1 - Qx\ where c is a real constant less than unity ; and write ac = 1, y Google 502 EXAMPLE OF A [162. SO that a is a, real constant greater than unity. Here, the points ic= 0, 1, tt, X are ea«h of them singular; and the value of a is ^ for each of them. Consequently, , l(i-i), Hi-i) , i(i-i) ,^ B c , '- -„'' '* (p-iy +(«-»,)■ + ». + ^- ! + «-«• and the conditions for « = co give One constant in / is left undetermined by these conditions ; thus ^="+(J^ + (S?S)'-^l)(»!-«)' say, where \ is the undetermined constant. U is possible to determine \, so that a: is a Fuchsian function of z, where z is the quotient of two solutions of the equation -- + /d = 0. dx' As regai-ds this Fuchsian function, its polygon may be obtained simply aa follows. We take four points A, B, G, D in the 2-plane to be the homologues of 0, 1, a, x ; owing to the value of a, which is ^ in each case, the internal angies of the polygon must each be \ir. We make the edges AB, CD conjugate, and likewise the edges BG, DA ; and then there is a single cycle, ADGB, the sum of the angles in which is %-Tr. With the former notation, we thus have 2 = 1, « = 2 ; so that 2p = 2 + 1 - 1 = 2, and therefore p = 1, as should be the case. Further, the sum of the angles of a curvilinear triangle, entirely on one side of the real axis, is less than tt, when the centres of the circular arcs he on the real axis : so that, if our polygon be curvilinear, the sum of its angles would be less than 27r (for it could be made up of two triangles), whereas the sum is actually 2Tr. Hence the polygon can only be a rectangle, and the Fuchsian functions are doubly- periodic. We therefore take a: = sn' ^, y^snzcnz dn z, y Google 62.] FUCHSIAN EQUATION s is manifestly permissible ; and then dx '2.y 2cM(a;-l)i(a.'-(i.)i' which leads to ''■■"I • L^ + (a, -1)' + (»■-(.)'] ^x{x^\){„-a) -21, so that we have X._J(„ + 1). This value of X renders x (and so y) a Fuehsian funetioa of the quotient of two solutions of the equation As regards the integrals of this equation, the indicial equation of« = Ois />(p-i) + iV-». SO that p = ^, p = I- Denoting by Vi and v^ the integrals that belong to J and J respectively, we have ! + §-:>■+... = ^-j;i . -i'-iii+c>e'+... = en" f, after the earlier analysis. Similarly, in the vicinity of « = 1, wo find integrals y Google 604 EXAMPLE OF A and then, taking ,^^. we find Now »-l- -cn= t = -(1- - -0- -«)(f-2)' + i(l-2»)(l-«)(f-^)' so that i, = (o-l)HI;-K). Hence !:-<-')' e-^)' so that, as where AB- -BC. - 1, because we have Again, i] nthe vicinity of « = a, we find integrals [162. cr,-(^-o)i|i + 2A;^j''^(^-») + ...|, and then, takini y Google 162.] FUCHSIAS EQUATION we find Also ic — cr = sn' c c = — dn^ f ^—-{K-K-iKj + }S^—^-—''hK-K-iKy^..., so that Proceeding as before, this leads to the relations Lastly, for large values of x, we have F,-«i|l-A(l+o)i+...l, r,-^|l-l(l + o)i+...}; and then, taking w. we find Now --6--|<l+a)f.'+.... 1 1 «-sn-r =csL"(f-iin = c(r-iir7-ic-(r-!'Jr')'(i+<»)4 y Google 506 INTEGRALS EXPRESSED AS [162. SO that Proceeding as before, this leads to the relations If, = ci (v^ - iK\) \ Wy = o-^v, ] ■ The relations, in fact, have enabled us to construct the expressions for each fundamental system in terms of the first and, therefore by inference, in terms of every other. Ex. 1. Discuss in the same way the Fuchsian differential equation 1 Ai f J 1 _\ 1 1^ connected with the eqiiation Ex. 2. Shew that, if whei'e p denotes Weieratrass's elliptic function, '•■ ■•'-•Ls^^lJ^ + (J:^' * (i:^'J"*(i-«,)(»-«,r(i^"S ■ and discuss the significance of the integral relation in regard to its paeudo- automorphic character for the equation Es:. 3. 4 f d e tal J hg x the s-plane is composed of two semi- circles, one upo a d am t i tl c real axis for values of s correaponding to values of a equal to and 1 the other upon a similar diameter for values of a: equal to 1 ■uid u (wl e e > 1) and of two straight lines drawn, through points corresj ondii^ to and a, perpendicular to the axis of real quantities. Prove that tlie subsidiary equation of the second order, for the construction of X as an automorphic function of the quotient of two of its integrals, is id,'- 'L"'^^ (»-!)■ + (i-o)>J^^»(«-l)(»-o)' where the constant jj is to be properly determined. AUTOMORI'HIC BY'NCTIONS USED TO MAKE THE InTEGRALH OF ANY Equation Uniform. 163. If, for any given equation, there is only one singularity, it can be made to lie at the origin. In order to obtain a variable s, in terms of which the integrals of the given equation can be expressed uniformly, we construct an y Google 163.] UNIFORM FUNCTIONS .507 equation of the second order which has i« = for a singularity, of such a form that the indicial ecjuation for x = has equal roots (§ 160). This auxiliary equation may have other singulari- ties, but otherwise it may be kept as simple as possible. Such an equation is the indicial ecjuation for »: = is e{d-l)=X. 90 that \~ — I if it has equal roots. Thus the equation is Two integrals are given by Vi = x^, fla = 3^ log « ; thU3 which is the new independent variable. An equation of the kind indicated ia {§ 45, Ex. 6) when the variable ia changed from :>: to s, where j; = e', the equation hccomea The integrals are synectic for all finite values of 3. 164, When a given differential equation has two singularities, a homographic transformation can be applied so as to fix them at X = 0. iK=l. To obtain a variable z in terms of which the integrals of the given equation can be expressed uniformly, we construct an equation of the second order, having and 1 as its singularities and such that the respective indicial equations have repeated roots. An appropriate equation is d'v _ a + ^w yGoosle 508 EXPRESSION OF INTEGKALS [164. The indicial equation for « = is p{p-l) = a, so that a= — ^ ; the indicial equation f or ic = 1 is p(p-l) = «+/3. SO that a + ^ = — ^, and therefore ,8 = 0, so that the equation is One integral is easily found to be and then z, the quotient of another integral by v,, is given by dz ^C ^ - 1 dx jii' x(a!~l)' on particularising the constant C, which may be arbitrary. Thus gives a new variable g, Huch that the integrals oi' the given differential equation arc uniform functions of s. Thus let the equatio: >a2' = <'. which has :f=0 and x=l for real singularitiea ; it is easy to verify that a:=ro ia not a Hingularity but only an ordinary point for every integral. When the equation is transformed so that b is the indepeodent variable, it becomes the integrals of which clearly are uniform functions of s. 165. When a given differential equation has three singulari- ties, a homograpfiic transformation can be used so as to fix them at a;=0, 1, oc . We may proceed in two ways. It may be possible to choose, as the fundamental region in the s-planc, a triangle, having circular arcs for its sides, and having Xtt, fi-rr, vtt for its internal angles at points which are the homologues of 0, oo , 1 respectively : y Google 165.] AS UNIFORM FUNCTIONS X, fi,, V being the reciprocals of integers. Then the equation may be taken in the form i(^% iO^-A^) iOr''') , i^'-'-'-^"'-' which is the norma! form of the equation of the hypergeometric series with parameters a, jS, 7, where X' = a-7)<, ^■ = (a-«', „■ = (,- = -/3)-. The variable s may be taken as the quotient of two solutions of the subsidiary equation ; and so .fCa + l-T, ff + 1- ") It is known* that x, thus defined, is a uniform automorphic function of s. This transformation will render uniform the integrals of a differential equation, which has no aiogularities except at 0, 1, 00 , provided the integrals are regular in the vicinity of those singularities and belong to indices which are integer multiples of X, V, fi respectively. If these conditions are not satisfied, in particular, if the singularities are essential for the integrals, then we proceed by an alternative methoi^. We take a subsidiary equation having 0, 1, co for singularities, such that the indicial equation for each of them has equal roots. Let it be where a', ^3", y are to be chosen so that the indicial ecjuation for each of the singularities has equal roots. These equations are p(p-l) = a', <T(a--l) = a' + ^' + y', t(t + 1) = 7', so that "'--i, f>' = i. 7'--i, and thus the equation is • r. F., § 275. y Google 510 GENERAL APPLICATION OF [165. The coefficient of v is the invariant of a hyp orgeo metric equation, of which the parameters are a = 0^l, 7 = 1; so that s, the quotient of two integrals v, is also the quotient of two integrals of the equation ''diu -i» = 0. This is the equation of the quarter-periods in elliptic functions: BO that This relation effectively defines a; as a modular function* of z : the fundamental region is a curvilinear triangle. The function exists over the whole s-plane : the axis of real quantities is a line of essential singularity. Any differential equation, having a; = 0, 1, v> for all its singu- larities no matter what their character may he, can be transformed by the preceding relation so that a is the independent variable ; its integrals are then expressible as functions of z which are uniform over the whole of the e-plane, their essential singularities lying on the axis of real quantities. Ea:. A differential equation haa only tliree singularities at x=a, 6, e, such that the roots of the indicial equations of those points are int<^er multipleB of a, 8, y respectively, where a, ft y are reciprocals of integers. Shew that a variable, in terms of which the integrals can bo expressed as uniform functions, ia given by taking the quotient of two Riemann P-functioiis with the appropriate singularities aiid indices. AuTOMORPHic Functions applied to General Linear Equations of any Order. 166. At the beginning of the preceding explanations and discussions, it was assumed {§ 157) that all the singular values of X are real. The assumption was then made for the sake of simplicity : it can be proved"!- to be unnecessary. * T. F., % 303. + Poincare, Ada Math., I. iv, pp. 246—250. y Google 1.66.] AUTOMORPHIC FUNCTIONS 511 Firstly, let the singularities be constituted by a,, a^, ..., Om, all of which are real, and by c, which will be supposed complex. With these we shall associate Co, the conjugate of c ; and we write 4>{x) = {x-c){a:-c„), a quadratic polynomial with real coefficients. Then all the quantities are real. Construct a fundamental region in the ^-plane, such that the foregoing m + 2 quantities are the homologues of the comere ; and let be the relation that gives the conformal representation of the region upon half the X-plane, so that F(s) is a Fuchsian faoction of 2. Consider the variable x, as defined by the equation So long as s remains within the fundamental region, a: is a uniform function of e; it could cease to be so, only if that is, if a! = ^c + ^c„, and then we should have y(») = .f(ic + }o.), which is not possible for values of z within the region. Also, J- is not zero for any value of s within the region; for then we should have which would make a zero magnification between the X-plane and the ^-region: this we know to be impossible for internal ^-points. This uniform function x, whose derivative does not vanish within the polygon, cannot acqiiirc either of the values c or Co within the polygon, for then we should havo F{s) = 0. yGoosle 512 AUTOMOBPHIC FUNCTIONS AND [166. which is possible only at a comer. Nor can it acquire any of the values til, eta, ..., a™ for points within the ;r- polygon : for at any such value, we have i*" («) = *(»). which again is possible only at a corner. Now since X = l'(3) is a relation that con formally represents the half X-plane upon a ^-polygon bounded by circular arcs (this polygon being otherwise apt for the construction of autornorphic functions), we have (§ 157) where ^ (X) is a rational function of X. But for any variables X and X, we have and therefore, in the present case, [z, «) = 2 (2a; ~ c - c„f f(af- cz -c^ + cc„) - = 2^(^), say, where '^V(x) is a rational function of ie. Hence s is the quotient of two integrals of the equation S + '^W'"- Now cc is known to be a uniform function of s ; it is therefore a Fuchsian function of z. And we have proved that, for values of z within the polygon, x cannot acquire any of the real values (ti, Oa, ..., (tm or either of the complex values c, d, and, further, that ^- does not vanish. as Secondly, to extend this result to the case, when x is not to acquire any one of any number of complex values for ir-points within the polygon, we adopt an inductive proof; we assume the result to hold when there are q — 1 pairs of conjugate complex values, and shall then prove it to hold when there are q pairs. It has been proved to hold, (i), when there are no complex values and, (ii), when there is a pair of conjugate complex values : it thus will be proved to hold generally. (2^ yGoosle 166.] LINEAR EQUATIONS IN GENERAL 513 Suppose, then, that the given ic-singnlarities arc made up of a number m of real values a,,a.i, ...,fflm>and of a number of complex values. Let the latter be increased in number by associating with each complex value its conjugate complex, whenever that conjugate does not occur in the aggregate ; and let the increased aggregate be denoted by arranged in conjugate pairs. Write which is a polynomial of ( equation fi-ee 2q with real coefficients. The dx = 0, of degree 1q~l with real coeOicients, certainly p root ; its other roots, when not real, can be arranged in conjuj pairs, the number of pairs not being greater than g — 1. Let its roots be denoted by h, h, ..., Vi. an aggregate which contains not more than 5 — 1 conjugate pairs. In the series of quantities 0; <f.(aO, ■-, 0(«-«); 0(M. ■■■. <^(6^-.); there are certainly m + 2 real quantities ; and there are not more than 5 — 1 conjugate pairs of complex quantities. According to our hypothesis, a Fuchsian function G{z) can be constructed, such that the foregoing m + 2 + 2 (5 - 1) quantities are the homologues of the comers of an appropriate fundamental region, and (?' (s) does not vanish within the region. Then, proceeding on the same lines as in the simpler case, we consider a variable .v, defined by the relation *W = (?{«). So long as s remains within the fundamental region, ic is a uniform function of 2 ; it could cease to be so, only if that is, it'a: = 6,, 6j or b;q-,, and then we should have e(«)-*(M, 4,(b,), .... or 4,(b„^o, y Google 514 FL'CHSIAN EQUATIONS HATING [lfi6. which is not possible for values of s within the region. Also, -J- does not vanish for values of z within the region ; for otherwise we should have for such values, and this is known not to be the ease. Further, m, being a uniform function of z whose derivative does not vanish for values within tbe polygon, cannot acquire any of the values c,- or Cf, for r=\, ...,q, within the polygon; if it could, we should have (i (as) = there, and then i'W-o, which is possible only at a corner. Nor can it acquire any of the values a,, ..., Om for values of ^ within the polygon ; if it could, we should have F{z) = 4,{<h), (f){a,), ..., or ^(0> which again is possible only at a comer. Now since Y, = G(z), is an automorphic function, it follows* that which is equal to — (e, Y], also is an automorphic function. Consider the upper half of the F-plane. So far as the equation Y = G(z) is concerned, certain points on the upper side of the axis of real quantities are exceptional, not more than g — 1 in number ; these can be considered as excluded, and cuts drawn from them to singular points on the real axis. We then can regard this simply-connected and I'esolved half- plane as conformally represented upon the polygon by the equation F = G (s) ; hence-f where 5(F) is a rational function of F. But where </> (x) is a polynomial ; hence 1^, «l-h Yli-^y + ir,-] y Google 166.] ASSIGNED SINGULARITIES 515 say, where 0(a;) is a rational function of x. Hence z is the quotient of two integrals of the equation Now a; is known to be a uniform function of z. It is therefore a Fuchsian function of z, which acquires the particular assigned values only at the corners of the fundamental region and nowhere within the region ; its derivative does not vanish anywhere within the region. The statement is thus established. 167. The preceding explanations, outlines of proofs, and analysis, will give an indication of the kind of result to be obtained, and the kind of application to differential equations to be made. It will be recognised that such proofs as have been adduced are not entii'ely complete : thus, when a number of real constants is to be determined by the same number of equations, whether algebraical or transcendental, it would be necessary to shew that the constants, if determined in the precise number, are real. As, however, it was stated at the beginning of these sections that only an introductory sketch of the theory would be given, there will be no attempt to complete the preceding proofs : we shall be content with referring the student, for the long and com- plicated processes needed to establish even the existence of certain results without evaluating their exact form, to the classical memoirs by Poincar^, and to the treatise by Fricke and Klein, which have already been quoted*. It may be convenient to recount the most important and central results of Poincar^'s investigations, which have any application to the theory of linear differential equations. Let be a linear equation of order g, having rational functions of x and y for its coefficients, where y is defined in terms of a; by the algebraic equation ' A memoi)- by E. T. Wliittaker, "On the eonneiion of algebraic funotions with automorphic functions," Fhil. Trami. (1899), pp. 1—32, may also be consulted. y Google 516 POINCAEfi'S THEOREMS [167. tbis equation in w will be called the main equation. Let da:' H'.y) be another equation, in which Six, y) is a rational function of w and y ; it will be called the subsidiary equation, and its elements are entifeiy at our disposal. Let cc = o,^, 2/ = &^,bea singularity of the main equation. If all the integrals are regular at this singularity, if they ai-e i'ree from logarithms, and if they belong to exponents, which are commensur- able quantities {no two being equal), let fc~' {where k is an integer) be a quantity such that the exponents are integer multiples of ^'. We make a; = a^, y = b^,& singularity of the subsidiary equation. In the case of the indicated hypothesis as feo the integrals of the main equation, we make the difference of the two roots of the indioial equation of the subsidiary equation equal to k~^. In every other case, we make those two roots equal. This is to be effected for each of the singularities of the main equation. Thus the subsidiary equation is made to possess all the singularities of the main equation. It may have other singulari- ties also ; for each of them, the difference of the two roots of the corresponding indicial equation is made either zero or the re- ciprocal of an integer, at our own choice. By these conditions, the coefficient 6(x,y) will be partly determinate: but a number of i will remain undetermined. The effect of these conditions is, by the analysis of § 160, to make x and y uniform functions of e, where z is the quotient of two linearly independent integrals of the subsidiary equation ; and no further conditions for this purpose need be imposed upon the parameters, which may therefore be used to secure other properties of the uniform functions. The various forms of 6, corresponding to the various determinations of the parameters, determine a corresponding number of differential equations ; all of these are said to belong to the same type, which thus is characterised by the singularities and their indicial ( Poincar^ has proved a number of propositions connected with e results that can be obtained by the appropriate assignment y Google 167.] ON AUTOMORPHIC FUNCTIONS 517 of values to these parameters. Of these, the most important I. It is possible to assign a iinique set of values in such a way as to secure that x and y are Fuchsian functions of z, existing only within a fundamental circle. II. It is possible to assign sets of values, unlimited in number, in such a way in each case as to secure that x and y are Kleinian functions of z, existing over only part of the 2-plane. III. It is possible to assign a unique set of values in such a way as to secure that x and y are Fuchsian functions or Kleinian functions of z, existing over the whole of the 3-plane. There are limiting cases when the Fuchsian function becomes doubly- periodic, or simply- periodic, or rational. PoiKCAR^'s Theorem that any Likbar Equation can be INTEGRATED BY" MEANS OF FuCHSUN AND ZeTAFUCHSIAN Functions. 168. Consider now the integrals of the main differential equation, when they are expressed in tenns of the variable z. We shall assume that x and y have been determined as Fuchsian functions oi z, existing only within the fundamental circle. Near an ordinary point x^, y„, any integral w is a holomorphic function oix — x„; near such a point, ic is a holomorphic function of 2 — 2o ; so that w, when expressed as a function of 2, is a holo- morphic function of z. In the vicinity of a singularity (a, b), there are two cases to consider. If all the exponents to which the integrals belong are commensurable quantities, so that they are integer multiples of some proper fraction &-^ where k is an integer, and if the integrals are free from logarithms, then every integral is of the form «. = (x-»)'S(^-«). y Google 518 POINCAR^'S THEOREM AS TO THE [168. where >5 is a holomorphic function of x — a,. As in § 160, we have SO that where T and li are holomorphic functions. Hence w = {z-cYG{z~c), where the function G is holomorphic in the vicinity of c. Thus w is a uniform function of 3 ; if /i is positive, then c is an ordinary point ; if ^ is negative, it is a pole. In all other cases, whether the integrals involve logarithms, or the exponents to which they belong are not all commensurable, or the singularity is one where some of the integrals, or even all the integrals, are irregular, the roots of the indicial equation for the subsidiary equation are equal. In consequence, the two circular arcs of any polygon touch, and thus the angular point is on the fundamental circle. As we consider the values of z within the fundamental circle, the character of the integral, when expressed as a function of z, does not arise for the point of the kind under consideration. It thus appears that, when z is restricted to lie within the fundamental circle of the Fuchsian functions which are the repre- sentative expressions of x and y, any integral of the main equation is a uniform function of z. When this uniform function has poles, it can be represented in the form where the zeros of G, (z) are the poles of the integral in unchanged multiplicity, and both 6 {s) and Qi {z) are holomorphic functions of z, within the fundamental circle. When the uniform function representing the integral has no poles, it can be expressed in the form where the function H (z) is holomorphic everywhere within the fundamental circle. Hence we have Poincare's theorem* that the integrals of a linear differential equation with algebraic coefficients can be ex- pressed as uniform functions of an appropriately chosen variable. * Acta Math., t. IV, p. 311. y Google 16!).] INTEGRALS OF LINEAR EQUATIOKS 519 169, The characteristic property of these uniform functions can be obtaiued as follows. Takiug the equation in the form where it is supposed that the term (if any) which involved ".-^^i has been removed from the equation by the usual substitution (§ 152), we denote by 0,, 0^, .... 8,, a fundamental system of integrals in the vicinity of any singularity (a^, i„). Let a closed path on the Eiemann surface, associated with the permanent equation, be described round the singularity; then, when the path is completed, the members of the fundamental system have acquired values ^i'. 6^, ..,, 6q, such that e^ = a["^l e, + al'^le^ + ... + a^^^l e^. {« = i, 2, . . . , 7), where the coefficients a*^' are constants such that their determ- inant is unity, because the derivative of order next to the highest is absent from the differential equation. Now ce and y are Fuchsian functions of z, existing only within the fundamental circle in the z-plane ; hence, when the path on the Riemann surface, which cannot be made evanescent, is com- pleted, ic and y return to their initial values, and z has described some path which is not evanescent. It follows, from the nature of the functions, that the end of the s-path is a point in another polygon, homologous with the initial position, so that the final position of z is of the form a^e + Q ^ 7.^ + K ■ The integrals dt, d„ ..., 6q are uniform functions of z\ let them be denoted by 4>i{z), <fh(^)i •■■, 4'q(^)- Moreover, B„' is the value of Sa «t tJie conclusion of the path ; thus "Wz + bJ' so that the integrals in the fundamental system consist of a set of uniform functions of z, which are characterised by the property y Google 520 ZETAFUCHSIA?J [169. Corresponding to the substitution of the Fuehsian group, we have a linear substitution S^ in the quantities 0i, 0^1 ■■-. 'f'g- the aggregate of these linear substitutious 8^, forms a group, which is isomorphic with the Fuchsian group. Functions of this pseudo-automorphic character are called* Zetafuchsian by Poincar^ : and thus we can say that linear differ- ential equations can be integrated by means of Fuchsian and Zeta- ^cksian functions which are uniform. It is, however, necessary to obtain explicit expressions for the functions 0, in order that the equation may be regarded as integrated, This is effected (I.e.) by Poiocar^ as follows. Let represent the substitution inverse to yS^, so that the quantities A^^ are the minors of the determinant of S^. Take any q arbitrary rational functions of z, say Hi{s), S^i^), -.., Hqip); and by means of them, in association with the Fuchsian group, con- struct p infinite series, defined by the equations ?,(.) = siA'''^ir.( s + hii'^i^+^if"" for the q values 1, ...,qoi fi.; the quantity m is a positive integer; and the summation with regard to i is over all the substitutions of the Fuchsian group. This integer ni is at our disposal : by choosing it sufficiently large, and by limiting the rational func- tions H, so that no one of the quantities is infinite on the fundamental circle, all the series can bo made absolutely converging: but we do not stay to establish this resulff. Assuming this convergence, and writing * Acta Math., t. v, p. 237. t It can be establishea on the same liaes as the convergence of Poincai^'s Thetafuchsian series: T. F., gg 304, 305. y Google 169.] FUNCTIONS 521 so that, for any value of k and al! the vahies of i, we get all the values of p for the group, we have But Owing to the properties of the isomorphic groups, we have and therefore SkS,'' = sr. that is, and therefore V7j;3 + 6i/ n=I l^" Now let 0(2) represent a Thetafuchsian series*, with the parametric integer m, and possessing the foregoing Kuchsian group: then, for each substitution of the group, we have ^(S^')^(«^+«~^w- We introduce functions Zi, Z^, ..., Z^, defined by the relations (f = l ?)■ They satisfy the conditions and therefore we may take or the q functions Z, which are Zetafuchaian functions, constitute a system of integrals of the differential equation. 170. As regards the Zetafuchsian functions thus constructed, it will be noted that the rational functions Hu ,.., Hq, which y Google 522 PR0PBBTIK8 OF A [170. enter into their construction, are arbitrary; so that an infinite number of Zetafuchsian functions can be formed, admitting a Fucbsian group 6 and the linear group (say G) isomorphic with G. Further, the Thetafuchsiaa series @(z) with the parametric integer m is any whatever; but, as we have f^ _ f V \-f' l'^^l±_§A __ ^ so that we may take where P (cc, y) is any uniform function of le and y. The simplest case occurs when P{ie,y) = \. Again, we have z C"^ "^ A\ "ZM')* ««2, W+...-K '*'/.<^)i "1?*^ + sj" and therefore 1 dZ. <!§-■ so that t. IV + /3A V7,. iT^:)=«^ dZ, ■0 i d^S ■■-C^ i that is. -zj"-^ z + 13. :)=cf + "Z {41 rf^, da; \.7( ,z + S, v. A, • are a Zetafuchsian system, admitting the Fuchsian group G and the isomorphic lineai- group G. The same property is possessed for all the derivatives of any order of the system Zi, Z,^, ..., Z^ with regard to x. y Google 170,] ZETAFUGHSIAN SYSTEM 523 Tliis property is used by Poincare to obtain the most general expression for a Zetafiichsian system, admitting the groups G and Q. Let it be T^, T,, ..., T^; and construct the matrix , dZ, dt-'Z, T, ^" *.■■ ■' d^^ ' , dZ, d'-'Z, T, d'-'Z, ■' d^' ' T, Denote by (— l)°'~'A,i_] the determinant obtained by cutting out the a'" column from the matrix : then, by a known property of determinants, we have AA + A.!|- + ,.. + A,_ S^-.''.- ues of ». Hence „ a. . A, iZ, ■'"^"a, " A, ■& A,_, di-^Z^ When s is subjected to any transformation of .the group 0, the quantities in any column in the matrix are subjected to the corresponding linear transformation of the group G ; so that each of the 5 + 1 determinants Ao, Aj, .... A^ is multiplied by the determinant of the linear transformation. Hence A, -h A^ is un- altered, that is, it is automorphic for the substitution of the group G\ and therefore, as this property is possessed for each substitution. A,- -;- A^ is automorphic for the group Q. Conse- quently, Ar -!- A, is a rational function of x and y, say ?=-^.-. (f-O, 1, . ..,?-!)! T,..FA + F,-- dx . + n for 11 = 1, 2, ..., q. This is Poincar^'s expression for the most general Zetafuchsian system, admitting the Fuchsian group G and the isomorphic linear group G. y Google 524 CONCLUDING [170. Wo can immediately verify that Z^, ..., Zg satisfy a linear differential equation, having coefficients that are rational in x and y. For d-iZ d^Z^ d'iZ^ da^ ' dx'i'' '"' da:^ ' are a Zetafuchsian system, admitting the Fuchsian group G and the isomorphic linear group G ; and therefore rational functions t^o, 1^, ..., i^j_i exist, such that holding for ail values of n. Thus Z^, ..., Zq are integrals of the linear differential equation Similarly, T„ ..., Tq are integra!s of a linear differential equation also of order q, having rational functions of x and y for its co- efficients, and characterised by the same groups G and G as characterise the equation satisfied hy Zi, ...,Zq. Concluding Remarks. 171, The Zetafuchsian and Thetafuchsian functions thus used occur, for the most part, in the form of series of a particular kind; as they vpere first devised by Poincar^, his name is fre- quently associated with them. The main aim in constructing them was to obtain functions which should exhibit, simply and clearly, the organic character of automorphism under the substi- tutions of the groups; and they are avowedly intended* to be distinct in nature from series adapted to numerical calculation, such as series in powers of z. Unless both these properties, viz. the exhibition of the organic chai'acter of the function and its adaptability to numerical calcu- lation, are possessed by the functions involved, it is manifest that they are not in the most useful form. It is unlikely that the best development of the general theory can be effected, until " Acta Math., t. v, p. 211. y Google 171.] REMARKS 625 functions have been obtained in a form that possesses both the properties indicateci. In this connection, Klein* quotes a parallel instance from the theory of elliptic functions, viz. the series of the form S S (mw + wi'ft)')"", usedf by Eisenstein, which exhibit the characteristic antomorphic property of the modular functions, but are not adapted to nu- merical calculation Their deficiency in this respect has been met by the po&seBsion of the theta-functions and the sigma- functions The generalisation of the Jacobian th eta-function and the Weierstrassian sigma- function, required for automorphic functionb, has not yet been attained. We thus letum to the statement made at the beginning of the foiegoing sketch of Poincar^'s theory of linear differential equations with algebraic coefficients. The explicit analysis con- nected with the theory of automorphic functions has not yet acquired sufficiently comprehensive forms upon which to work ; and therefore its application to linear differential equations, as to any other subject, can be only partial and imperfect in its present st^;e. The theory of automorphic functions in general presents great possibilities of research : the gradual realisation of these possibilities will be followed by corresponding developments in many regions of analysis. • Vorlesungen S. lineare Differentialgleiehungen d. sweiten Ordnting, (Gottingen, 1894}, p. 496. See also Fricke imd Klein, Theorie der automoTpken Functionen, t. II, p. 156. + For references, see T.F.,% 56. y Google y Google INDEX TO PART III. I. er ed g 2'i ' M- ed bl 54 t tl 257. gul Algebraic coeMeieDts, ei^uatiotis having, Chapter x; oharnoter of integrals of, in vioinity of branch-point, 479, and in vicinity of a singularity, 480 ; mode of constructing integrals of, 483 ; Appell's class of, 484 ; aseocisited with automorphio fnnetiona, 488 (see auto- morphic fmictioas). Algebraio equation, roots of, satisfy a linear equation with rational ooefii- oienta, 46, 174 ; connected with differ- ential resolvents, 49. Algebraic integrals, eq.uations having, 45, 165, Chapter v ; oonneoted with theory of finite groups, 175 ; connected with theory of oovariants, 175; equa- tions of second order having, 176 et Beq. ; equations of thin] order having, 191 et seq. ; eqaationa of fourth order having, 201; constraction of, 184, 198 ; and homogeneons forms, 202. Aji^ytieid form of group of integrals associated with multiple root of funda- mental equation of a singularity, 66 ; likewise for multiple root of funda- mental equation for a period or periods, 416, 454. Anuulus, integral converging in any (see fundamental equation, irregular inte- gral, Lawent seriet). tmormaJes, 270. Apparent singularity, 117; conilitLons for, 119. Appell, 209, 484. tl I 117 A t m pb t t d d ft t al equationshavingalgebiaic coefficients, 48S ; and confonnal representation, 491 ; associated with linear equations of second order, 495 ; constructed for a speoid case, 600 ; when there is one singularity, 6iD6; when there are two singularities, 508 ; when there are three, 509, 610; in general, 510 et Barnes, 448. begleitender bilinearer Differentialaua- dritek, 254. Benoit, 474. Bessel'E equation, 1, 13, 84, 100, 101, 126, 164, 330, 333, 393. Bilinear ooDcomitant of two reciprocally adjoint equations, 254. Bdoher, 161, 169. B6tlier's theorem on equations of Fuchs- ian type with five singularities, 161. Boole, 229. Boulanger, 195, 197, 198. BrioscM, 206, a08, 218. Casorati, 55, 60, 417. Caochy, 11, 30. Canohy'a theorem used to establish existence of syneotic integral of a linear eq nation, 11. Cayley, 94, 113, 121, 182, 216, 233, 246, y Google 528 INDEX TO PAET 111 Cels, 2S4. CharaoleriBtiG equation belonging to a singalurity, 40. Churacteristic equation toe deteimming fitctor of normal integrals, 294 ; effect of simple root of, 294; effect o£ multiple root, 398. Cbaracterietic function of an equation, Cbaracteristic index, definerl, 221 ; and namlier of regulav integrals of an equation, 930, 233 ; of reciprooally adjoint equatione, the eame, 257. ChrjBtal, 7, 83. Circular cjlinder, differential equation of, 164 ; (see BeseeVi equation). Class, equations of Fuchsian (see Ji'tic/is- ian type). Cockle, 49. CoefHcients, form of, near a eingularit; if all integtalB there are regular, 78. Collet, 20. Oonformal representation, and auto- morphic functions, 491 ; ami funda- mental polygon, 493. Constant ooefBoients, equation having, 14—20. Construction, of regular integrals, liy method of Frobeniua, 78; of normal integrals, periodio integrals (see normal integrals, simply-periodie iniegraU, doubly-penodic integraU). Continnation prooeae applied to ftyneetio integral, 20. Continaed fractions used to obtain a fundamental equation, 439. Covariants associated viitli algebraic integrals, 202 ; for equations of third order, 203, 209; for equations of fourth order, 204 ; for equations of second order, 206. Craig, vi, 411. Crawford, 474. Curve, integral, defined, 303, HOS. Darbous, 20, 254, 475. Definite integrals (see Laplace's definite integral, double-loop integral). Determinant of a system of integrals, 25 ; its value, 37 ; not vanishing, the system is fun- damental , 29 ; of a fundamentaJ system does not vanish, 30 ; il form of, for one particular syst , 34; form of, near a. aingalarity, 77 ; when the ooeffioienta are peri- odic, 406, 446 ; when the co- efB.cients are algebraic, 481. Determinants, infinite (see infinite de- tertainanta). Determining factor, of normal integral, 262; obtained by Thome's meHiod, 262 et seq.; conditions for, 265; tor ' itegrals of Hamburger's equations, Diagonal of infinite determinant, 349. Difference-relations, 63, 417. Diffei-ential invariants (see invariants, differential). Differential resolvents, 49. Dini, 254, 256. Divisors, elementary (see elementary Double-loop integrals, integi'ate equations, mi ei seq. Doubly-periodic ooefBoienta, eqnaiiona having, 441 et seq. ; substitutions for the periods, 413 ; ^ndamental equa- tions for the periods, 444, 445. Doubly-periodio integrals of second kind, 447 ; Ficaid's theorem on, 447 ; num- ber of, 448, 450 ; belonging to Lamp's equation, 463; how constructed, 471, 475. applied t Blement of fundamental system. 30. Elementary divisors, of certain determ- inants, 41 — 43 ; of the fundamental equation, 65; determine groups and sub-groups of integrals. 62 : effect of, upon number of periodio integrals when coefficients are periodio, 416, 460. Elliott, M„ 434, 425, 474. Elliptic cylinder, differential equation of, 164, 399, 431—441. Expansion of converging infinite de- terminants, 363. Expansions, asymptotic (see asymptotic expamioiiK). Exponent, to which regular integral belongs, 74; properties of, 75; to which the det«rminant of a fundamental system belongs, 77; to which normal integral belongs, 262; of irregular integral as zero of an infinite determinant, 368, Exponents, snm of, for equations of Fuchsian type, 128 ; for Bieraann'a P-fnnotion, 139. Fabry, 94, 270. Factor, determining (see determining Jaetoi-). Fano, 214, 218. Finite groups of lineo-linear substitu- tions, in one variable, 176; connected with polyhedral functions, 181 ; asso- ciated with equations of second order having algebraic int^rals, 182 ; used y Google INDEX TO PART III 529 for con struct ion of algebra 185; Integra J vftriablea, 193 ; their dif- ferential invariants, 195; the Laguerre invariaot, 196 ; used to construct equations of third order having algebraic integrals, 197; in three variables, 300. First kind, periodic function of, 410. Floqnet, 231. 334, 259, 411, 418. Frioke, 489. 515, 535. Frobenius, 78. 93, 109, 336, 231, 233, 338, 247, 254, 257, 369. Frobffliius' metliod, for the construction of integrals (ail being regular), 79 et aeq. ; variation of, suggested bj Cajley for some cases, 114; applied to hyper- geometi^c equation for special cases, 147; for the construction of integrals, when only some are regolar, 335 et seq.; used for construction of irregular integrals, 379 et seq. Fuchs, L., 10, 11. 60, 65, 66, 79, 93, 94, 109, 110, 117, 133, 135. 126, 129, 15S, 306, 208, 216, 399, 462. Fnchsian equations, 123, 495 et seq. ; in- dependent variable a uniform function of quotient of integrals of. 496—199 ; mode of determining coefficients in. 501; used as subsidiary to linearequa- tions of any order, 515. Fnohsion functions, associated with linear equations of the second order, 500, 502, 516 ; associated with linear equations of general order. 615, 617; in the expression of integrals as uni- form functions, 620. Fnchsian group, {see Fuchsiaii /miction, Zetafuchsiam fanetioii). Focbsian type, equations of. Chapter iv, pp. 133 et seq.; form of, 123; proper- ties of exponents, 126 ; when fully determined by singu- larities and exponents, 128; of second order with any number of singularities, 150; forms of, when fc is an ordinary point, 152; when as is a singu- larity, 155, 158; Klein's normal, 158; Lamp's equation transformed so as to be of, 160; equations of. haviag live singu- larities, 161; Bdcher's theorem Fundamental equation, belonging to a singularity, is same for all fuuda- mental systems, 38—40; invariants of, 10; Poincar^'s theorem on, 40; properties of, connected with ele- mentary divisors, 41—43; fundamental system of integrals assnoiated with 50 whe i roots are simple 52 vhen a root is multiple 53 loots of how lelated to roots of mdioial equation 94 Fundamental equation when integrals are iriegular expressed as an infinite deteimmant i(89 luite terms 392 I methods of obtaining 399 Fundamental equations for double pen ods 441 145 their form 447 loots of determmedoubly peiioiio integrals of the second Mud, 448 ; number of these integrals, 150 ; efCeot of multiple roots of, 451. Fundamental equation for simple period, 406; is invariantive, 406; form of, 407 ; integral associated with a simple root, 408; integrals associated with a multiple root, 108; analytical expres- sion of, 419. Fundamental equation when coefficients are algebraic, 492; relation to in dicial equation. 482. Fundamental polygon for automorphic functions, 490, 493, 500. Fnndamental system of in tegrals, defined. 30; its deteiininant is not evanescent, 30; properties of , 30, 31 ; tests for,31, 32; form of, near singularity, 50; if root of fundamental equation is sim- ple, 62; if root is multiple, 53; affected by elementary divisors of fundamental equation, 67; aggregate of groups associated with roots of indicia! equation make fundamental system. 95. Fundamental system, of irregular inte- grals, 387; of integrals when coelfioi- ents are simply-periodic, 408, 419; when ooeflioients are doubly-jwriodio, 449 — 457; when coefGeients are alge- braic, 180. Fundamental system, constituted by group of integrals belonging to a multiple root of fundamental equa- tion (see grou,^ of integraU). Gordan, 183. Grade of normal integral, 269. Graf, 333. GreenhUl, 466. Glroup of tg 1 as led with mul- tiple t f f 1 tal equation, 53 ; r sol d t b g oups, by ele- mentary d 57 Hamborger's sub-g p f 62 g al analytical form t 5 1 fundamental systen tit f 1 'er order, 72. 34 yGoosle 530 INDEX TO PAET III Group of iotegralB, associated with mul- tiple root of iudioial eqaation in methocl of Frobeaiae, 80; general theorem on, 93; aggregate of , inal^e a fundamentSil aystem, 96 ; compared with Ham- burger's groups, 113. Group of integrals for hjpergeometrio equation, 144. Group of irregular integrals associated with multiple root of characteristic infinite determinant. 381 et seq.; re- solved into sub-groups, 382. Group of integrals associated with mul- tiple roots of fundamental equationsfoi' periods when ooefficlents are doubly- periodic, 451 ; analytical expression of, 482, 467; fnriiier development of, when uniform, 459. Group of integrals associated with mul- tiple root of fundamental equation for period when coefGcients are simply- periodic, 415 ; arranged in sub-groups, according to elementary divisors, 416; analytioi expression of, 419; they constitute a fondamental system for equation of lower order, 420 ; further expression of, when uniform, 421. Groups of substitutions, finite (see^mi(e groups); infinite (see automorphic ftinc- Griinfeba, 359. Gubler, 333. Gtinthev, 11, 299, 399. GjldSu, 469. Halphen, 254, 356, 281, 316, 316, 448, 464, 465, 473. Hamburger, 38, 60, 62, 63, 64, 113, 977, 380, 383, 286, 399, 489. Hamburger's equations, 276 et seq. ; of second order with normal ini grals, 979; the number of nc mai integrals, 280; ot general order n witii normal subnormal integrals, 288 et seq of third order with normal or sod- normal integrals, 301 et seq. Hamburger's sub-groups of integrals (seo sub-groups of integrals j. Hankel, 103, 333. Harley, 49. Heffter, 56, 156. Heine, 164, 166, 431, 441. Hermits, 15, 30, 448, 463, 465, 468, 473. Hermite, on equation with constant ooefUcienta, 15 — 20 ; on equation with doubly-periodic ooeffioients, 465. 50. Hiil, G. W., 348, 38 Hobson, 334, 337. Homogeneous forms {i 398, 399, 403, 482. linear equations, defined, 3; discussion limited to, 3. Homogeneous relations between inte- grals when they are algebraic, 203, 217; of second degree for equations of third order, 230; and of higher degree, 214. Horn, 333, 341, 342, 346, 347. Humbert, 167. Hypergeometrio function, used to render integrals of differential equations uni- form in speoial ease, 509. Hypergeometrio aeries, equation of, 1, 13, 34, 103. 136, 144—160, 173, 338, 601, 509. Identical relations, polynomial in powers of a logarithm, cannot exist, 69. Index, characteristic (see efiaraclei-UUe index) ; to which regular integral be- longs, 74; properties of, 75. Indiojal eqaation, when all integrals are regular, 85, 94; significance of, in the method of Frobeuius, 85 ; integral associated with a simple root of, 86; group of integrals associated with a multiple root of, 86; roots ot, how connected with roots of fundamental equation, 94; for equ t on w th not all integrals reg a 223 2''7 ludioial f n t regular J4 wh n not al ntegrals are regular, 227, degree of, as affecting the number of regular intf^Is, 230, 933 ; of adjoint equation, as affecting the number of regular integrals, 959. Infinite determinant, giving exponent of irregular integral, 368; modified to another determinant, 369; is a peri- odic function of its parameter, 375 ; effect of simple root of, 380. of a mul- tiple root of, 381 et seq. ; leads to the fundamental equation of the singu- larity, 389 ; expressed in finite terms, 392. Infinite determinants in general, 349; convergence of, 360 ; properties of con- verging, in general, 353 et seq.; uni- form convergence of, wben functions of a parameter, 358 ; may be capa- ble of differentiation, 359; used to solve an unlimited number of linear equations, 360 ; applied to construct irregular integrals of differential equa- y Google INDEX TO PART III 531 Initial oondJtiouB defined, 4 values, 4 ; efCeol of, apon form of synectic integral, 9 Integral curve, 203, 20S, lot^rals. irregnkr (see i g la ate grali). Integrals, doubly-periodic irregular normal, regular, simply-pe o!o b norma], ayneotio (see unde these t ties respeetivdy). Intep^ rendered uuifona f net ons of a variable, wben there s one singu tarily, oOG ; when there are tno s Dgu laritias, 608 ; when there are th ee 509, SIO; in general, 510 t sei by 1 of Z tat oh i n f ct ona 518 520 I 195 f I 1 'f h quat f i f tl thi d fth f th 01 213 Laguerre'e, 196. Invariants of fundamental equation, oon- neoted with singnlarity, 38, 40; for irregular integrals, 398; connected with a period or periods 405 i45 h fli t Ig b il bl p th q t th d f Lab ange 251 I ague e 196 Lan^ eiuaton 1 12b 151 IbO 16o 168 338 148 4b4— 473 IJam^ a generahsed equation IbO Laplace a defin te ntegral aat sf Tng equat an vith ational ooefhc enta 318 oonto r of 23 developed nto '.B here liie e exist 324 e eq isp eas ng an reg lar ntegral 364 proof of convergence w thin BJ1 annnlne 3b6 I/egeudre s equation 1 13 34 lOi 1 6 IbO 163 Liapounofl, 319, 42S — 431. Liapounoft'a theorem, applied to evaluate Laplace's definite integral, 324; me- thod of discussing uniform periodic integrals, 425. Lindemann, 4iU, 434, 437. Lindstedt, 439. Linear algebraic equations, infinite Eyetem of, solved by meana of infinite determinants, 360. Linear difierential equs^tion, definition of, a. Lineo-linear substitutiona (aee finite L g , quantity affected by, can uniform linear differential t and determine ita funda- tal atem, 66; d tical relatione, polynomial in fe lac iniegj^s free from, 106 j CO dition that some regular egral shall be free from, tai ed by g 1 sat t F b m th 1 379 th n.t t t Id M d 11 333. t 1 J tem 387 M k ft 19. bl bym'^ ft M f nfinite determinants. 354. ph f t (ae in. pi M tt g Leffler, 399, 463. f ) M d 1 f notion, naed to reader inte- 8 1 f differential eqaationa uniform da 197 200 Sii 334 33 341 p al case, 610; Eiaenatain'a g e 113 f aimUac to, 525. M It 1 1 oot, group of integrala asso- I in 150 153 15 15 161 17b 1 S tel th (see mi.itiyie root). 1S7 190 117 tb 4 9 515 2 M Itipli f periodic integral of second 1 m I f rm f eq t f ki d 410; ia ft root of the funda- d d dF h typ 158 m tal quation of the period, 406. m thod f eq ti f sec d M th 4 d h m Ig b tefe 1 176 N m 1 f m, (after Frobenius) of equa- 341 t h ing some inlegrala regular. K h d4 35J 398 3')9 4 3 27 1 component factors of such mm 14b eq tion, and of a composite ( Iter Kieia) of equation of Fuohs- y Google y Google